99 Sentences With "unital" | Random Sentence Generator

If A and B are two unital algebras, then an algebra homomorphism F:A\rightarrow B is said to be unital if it maps the unity of A to the unity of B. Often the words "algebra homomorphism" are actually used to mean "unital algebra homomorphism", in which case non-unital algebra homomorphisms are excluded. A unital algebra homomorphism is a ring homomorphism.

The 2×2-matrices over the reals form a unital algebra in the obvious way. The 2×2-matrices for which all entries are zero, except for the first one on the diagonal, form a subalgebra. It is also unital, but it is not a unital subalgebra.

Only for unital algebras is there a stronger notion, of unital subalgebra, for which it is also required that the unit of the subalgebra be the unit of the bigger algebra.

In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint functors page); there is no essential difference between modules for the resulting unital ring, in which the identity acts by the identity mapping, and representations of the algebra.

Hypercomplex number is a term for an element of a unital algebra over the field of real numbers.

Given two unital algebras A and B, an algebra homomorphism :f : A → B is unital if it maps the identity element of A to the identity element of B. If the associative algebra A over the field K is not unital, one can adjoin an identity element as follows: take as underlying K-vector space and define multiplication ∗ by :(x,r) ∗ (y,s) = (xy + sx + ry, rs) for x,y in A and r,s in K. Then ∗ is an associative operation with identity element (0,1). The old algebra A is contained in the new one, and in fact is the "most general" unital algebra containing A, in the sense of universal constructions.

In some areas of mathematics this assumption is not made, and we will call such structures non-unital associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital. Many authors consider the more general concept of an associative algebra over a commutative ring R, instead of a field: An R-algebra is an R-module with an associative R-bilinear binary operation, which also contains a multiplicative identity. For examples of this concept, if S is any ring with center C, then S is an associative C-algebra.

It is still unknown if this unital can be embedded in a projective plane of order 36, if such a plane exists.

However, one section is devoted to the theory of quasiregularity in non-unital rings, which constitutes an important aspect of noncommutative ring theory.

A unit in an allegory is an object for which the identity is the largest morphism U\to U, and such that from every other object, there is an entire relation to . An allegory with a unit is called unital. Given a tabular allegory , the category is a regular category (it has a terminal object) if and only if is unital.

Some authors use the term "associative algebra" to refer to structures which do not necessarily have a multiplicative identity, and hence consider homomorphisms which are not necessarily unital. One example of a non- unital associative algebra is given by the set of all functions f: R → R whose limit as x nears infinity is zero. Another example is the vector space of continuous periodic functions, together with the convolution product.

In this case of embedded unitals, every line of the plane intersects the unital in either 1 or n + 1 points. In the Desarguesian planes, PG(2,q2), the classical examples of unitals are given by nondegenerate Hermitian curves. There are also many non-classical examples. The first and the only known unital with non prime power parameters, n=6, was constructed by Bhaskar Bagchi and Sunanda Bagchi.

Unital composition algebras are called Hurwitz algebras. If the ground field is the field of real numbers and is positive-definite, then is called a Euclidean Hurwitz algebra.

In doing so one moves from fields to commutative superalgebras and from vector spaces to modules. :In this article, all superalgebras are assumed be associative and unital unless stated otherwise.

All examples in this section involve associative operators, thus we shall use the terms left/right inverse for the unital magma-based definition, and quasi-inverse for its more general version.

Polynomials can be evaluated by specializing the indeterminate to be a real number. More precisely, for any given real number there is a unique unital ring homomorphism such that ., §IV.1 No. 3.

Since Γ acts doubly transitively on P, this will be a 2-design with parameters 2-(q3 \+ 1, q + 1, 1) called a Ree unital. Lüneburg also showed that the Ree unitals can not be embedded in projective planes of order q2 (Desarguesian or not) such that the automorphism group Γ is induced by a collineation group of the plane. For q = 3, Grüning proved that a Ree unital can not be embedded in any projective plane of order 9.

Precisely, one has: :Nakayama's lemma: Let U be a finitely generated right module over a (unital) ring R. If U is a non-zero module, then U·J(R) is a proper submodule of U.

This is called the evaluation homomorphism. Because it is a unital homomorphism, we have . That is, for all specializations of to a real number (including zero). This perspective is significant for many polynomial identities appearing in combinatorics.

In the category of unital rings, there are no kernels in the category-theoretic sense; indeed, this category does not even have zero morphisms. Nevertheless, there is still a notion of kernel studied in ring theory that corresponds to kernels in the category of non-unital rings. In the category of pointed topological spaces, if f : X → Y is a continuous pointed map, then the preimage of the distinguished point, K, is a subspace of X. The inclusion map of K into X is the categorical kernel of f.

In geometry, a unital is a set of n3 \+ 1 points arranged into subsets of size n + 1 so that every pair of distinct points of the set are contained in exactly one subset. n ≥ 3 is required by some authors to avoid small exceptional cases. This is equivalent to saying that a unital is a 2-(n3 \+ 1, n + 1, 1) block design. Some unitals may be embedded in a projective plane of order n2 (the subsets of the design become sets of collinear points in the projective plane).

Every unital ring may be viewed as an additive category with a single object, and the quotient of additive categories defined above coincides in this case with the notion of a quotient ring modulo a two-sided ideal.

Another important and motivating example of a free ideal ring are the free associative (unital) k-algebras for division rings k, also called non-commutative polynomial rings . Semifirs have invariant basis number and every semifir is a Sylvester domain.

Let A be a finite-dimensional complex unital Jordan algebra which is semisimple, i.e. the trace form Tr L(ab) is non-degenerate. Let be the quotient of by the equivalence relation. Let be the subset of X of classes .

In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory.

Let R be a fixed commutative ring (so R could be a field). An associative R-algebra (or more simply, an R-algebra) is an additive abelian group A which has the structure of both a ring and an R-module in such a way that the scalar multiplication satisfies :r\cdot(xy) = (r\cdot x)y = x(r\cdot y) for all r ∈ R and x, y ∈ A. Furthermore, A is assumed to be unital, which is to say it contains an element 1 such that :1 x = x = x 1 for all x ∈ A. Note that such an element 1 is necessarily unique. In other words, A is an R-module together with a R-bilinear binary operation A × A → A that is associative, and has an identity. Technical note: the multiplicative identity is a datum (there is the forgetful functor from the category of unital associative algebras to the category of possibly non-unital associative algebras) while associativity is a property.

Any vector space can be made into a unital associative algebra, called functional-theoretic algebra, by defining products in terms of two linear functionals. In general, it is a non-commutative algebra. It becomes commutative when the two functionals are the same.

In geometric algebra, the outermorphism of a linear function between vector spaces is a natural extension of the map to arbitrary multivectors. It is the unique unital algebra homomorphism of exterior algebras whose restriction to the vector spaces is the original function.

50 The notion of a left quasiregular element is defined in an analogous manner. The element y is sometimes referred to as a right quasi-inverse of x.Polcino & Sehgal (2002), [ p. 298]. If the ring is unital, this definition quasiregularity coincides with that given above.

A bistochastic quantum channel is a quantum channel \Phi(\rho) which is unital,John A. Holbrook, David W. Kribs, and Raymond Laflamme. "Noiseless Subsystems and the Structure of the Commutant in Quantum Error Correction." Quantum Information Processing. Volume 2, Number 5, p. 381-419.

Let A be a finite-dimensional unital Jordan algebra over a field k of characteristic ≠ 2.In the main application in , A is finite dimensional. In that case invertibility of operators on A is equivalent to injectivity or surjectivity. The general case is treated in and .

In mathematical representation theory, the Hecke algebra of a pair (g,K) is an algebra with an approximate identity, whose approximately unital modules are the same as K-finite representations of the pairs (g,K). Here K is a compact subgroup of a Lie group with Lie algebra g.

This is a glossary for the terminology in a mathematical field of functional analysis. See also: List of functional analysis topics. Throughout the article, unless stated otherwise, the base field of a vector space is the field of real numbers or that of complex numbers. Algebras are not assumed to be unital.

180 Intuitively, quasiregularity captures what it means for an element of a ring to be "bad"; that is, have undesirable properties.Isaacs, p. 179 Although a "bad element" is necessarily quasiregular, quasiregular elements need not be "bad", in a rather vague sense. In this article, we primarily concern ourselves with the notion of quasiregularity for unital rings.

If A is a unital associative algebra over K with multiplication × then a quadratic map Q can be defined from A to EndK(A) by Q(a) : b ↦ a × b × a. This defines a quadratic Jordan algebra structure on A. A quadratic Jordan algebra is special if it is isomorphic to a subalgebra of such an algebra, otherwise exceptional.

The term "rng" was coined to alleviate this ambiguity when people want to refer explicitly to a ring without the axiom of multiplicative identity. A number of algebras of functions considered in analysis are not unital, for instance the algebra of functions decreasing to zero at infinity, especially those with compact support on some (non-compact) space.

Let and be commutative rings and be a ring homomorphism. An important example is for a field and a unital algebra over (such as the coordinate ring of an affine variety). Kähler differentials formalize the observation that the derivatives of polynomials are again polynomial. In this sense, differentiation is a notion which can be expressed in purely algebraic terms.

Other common names for the least element are bottom and zero (0). The dual notion, the empty lower bound, is the greatest element, top, or unit (1). Posets that have a bottom are sometimes called pointed, while posets with a top are called unital or topped. An order that has both a least and a greatest element is bounded.

This ring is an R-algebra, associative and unital with identity element given by .Kassel (1995), [ p. 32]. where 1A and 1B are the identity elements of A and B. If A and B are commutative, then the tensor product is commutative as well. The tensor product turns the category of R-algebras into a symmetric monoidal category.

The power series definition of the exponential function makes sense for square matrices (for which the function is called the matrix exponential) and more generally in any unital Banach algebra . In this setting, , and is invertible with inverse for any in . If , then , but this identity can fail for noncommuting and . Some alternative definitions lead to the same function.

In the mathematical field of representation theory, a trivial representation is a representation of a group G on which all elements of G act as the identity mapping of V. A trivial representation of an associative or Lie algebra is a (Lie) algebra representation for which all elements of the algebra act as the zero linear map (endomorphism) which sends every element of V to the zero vector. For any group or Lie algebra, an irreducible trivial representation always exists over any field, and is one-dimensional, hence unique up to isomorphism. The same is true for associative algebras unless one restricts attention to unital algebras and unital representations. Although the trivial representation is constructed in such a way as to make its properties seem tautologous, it is a fundamental object of the theory.

The fourth follows because :. In fact is quasi-invertible if and only if is quasi- invertible in the mutation . Since this mutation might not necessarily unital this means that when an identity is adjoint becomes invertible in . This condition can be expressed as follows without mentioning the mutation or homotope: In fact if is quasi-invertible, then satisfies the first identity by definition.

Nathan Jacobson described the automorphisms of composition algebras in 1958. The classical composition algebras over and are unital algebras. Composition algebras without a multiplicative identity were found by H.P. Petersson (Petersson algebras) and Susumu Okubo (Okubo algebras) and others.Max-Albert Knus, Alexander Merkurjev, Markus Rost, Jean-Pierre Tignol (1998) "Composition and Triality", chapter 8 in The Book of Involutions, pp.

The inner automorphisms form a normal subgroup of Aut(G), denoted by Inn(G); this is called Goursat's lemma. The other automorphisms are called outer automorphisms. The quotient group is usually denoted by Out(G); the non-trivial elements are the cosets that contain the outer automorphisms. The same definition holds in any unital ring or algebra where a is any invertible element.

Let be a Hilbert space and the bounded operators on . Consider a self-adjoint unital subalgebra of (this means that contains the adjoints of its members, and the identity operator on ). The theorem is equivalent to the combination of the following three statements: :(i) :(ii) :(iii) where the and subscripts stand for closures in the weak and strong operator topologies, respectively.

The main technique used in this proof, namely rewriting the channel of interest as a convex combination of other simpler channels, is a generalization of the method used earlier to prove similar results for unital qubit channels.C. King, Additivity for unital qubit channels The fact that the classical capacity of the depolarizing channel is equal to the Holevo information of the channel means that we can't really use quantum effects such as entanglement to improve the transmission rate of classical information. In this sense, the depolarizing channel can be treated as a classical channel. However the fact that the additivity of Holevo information doesn't hold in general proposes some areas of future work, namely finding channels that violates the additivity, in other words, channels that can exploit quantum effects to improve the classical capacity beyond its Holevo information.

They call an element x right quasiregular if there exists y such that x + y + xy = 0,Kaplansky, p. 85 which is equivalent to saying that 1 + x has a right inverse when the ring is unital. If we write x\circ y=x+y+xy, then (-x)\circ(-y)=-(x\cdot y), so we can easily go from one set-up to the other by changing signs.Lam, p.

As noted above, the results of PCA depend on the scaling of the variables. This can be cured by scaling each feature by its standard deviation, so that one ends up with dimensionless features with unital variance.Leznik, M; Tofallis, C. 2005 Estimating Invariant Principal Components Using Diagonal Regression. The applicability of PCA as described above is limited by certain (tacit) assumptionsJonathon Shlens, A Tutorial on Principal Component Analysis.

The free non- associative algebra on a set X over a field K is defined as the algebra with basis consisting of all non-associative monomials, finite formal products of elements of X retaining parentheses. The product of monomials u, v is just (u)(v). The algebra is unital if one takes the empty product as a monomial. Kurosh proved that every subalgebra of a free non-associative algebra is free.

Everything else proceeds as described above: upon completion, one has a unital associative algebra; one can take a quotient in either of the two ways described above. The above is exactly how the universal enveloping algebra for Lie superalgebras is constructed. One need only to carefully keep track of the sign, when permuting elements. In this case, the (anti-)commutator of the superalgebra lifts to an (anti-)commuting Poisson bracket.

In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing.

The unitary case can only occur if is a square; the absolute points and absolute lines form a unital. In the general projective plane case where duality means plane duality, the definitions of polarity, absolute elements, pole and polar remain the same. Let denote a projective plane of order . Counting arguments can establish that for a polarity of : The number of non-absolute points (lines) incident with a non-absolute line (point) is even.

The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by giving an antipode to create a Hopf algebra structure. Note: In this article, all algebras are assumed to be unital and associative. The unit is explicitly required to define the coproduct.

Hamiltonian systems can be generalized in various ways. Instead of simply looking at the algebra of smooth functions over a symplectic manifold, Hamiltonian mechanics can be formulated on general commutative unital real Poisson algebras. A state is a continuous linear functional on the Poisson algebra (equipped with some suitable topology) such that for any element of the algebra, maps to a nonnegative real number. A further generalization is given by Nambu dynamics.

Let be a vector space over the field . Informally, multiplication in is performed by manipulating symbols and imposing a distributive law, an associative law, and using the identity for . Formally, is the "most general" algebra in which these rules hold for the multiplication, in the sense that any unital associative -algebra containing with alternating multiplication on must contain a homomorphic image of . In other words, the exterior algebra has the following universal property:See , and .

This implies in particular that is compact. There is a more direct proof of compactness using symmetry groups. Given a Jordan frame (ei) in E, for every a in A there is a k in U = Γu(A) such that with (and if a is invertible). In fact, if (a,b) is in X then it is equivalent to k(c,d) with c and d in the unital Jordan subalgebra , which is the complexification of .

In linear algebra, the dual numbers extend the real numbers by adjoining one new element (epsilon) with the property ( is nilpotent). Thus the multiplication of dual numbers is given by : (a+b\varepsilon)(c+d\varepsilon) = ac + (ad+bc)\varepsilon (and addition is done componentwise). The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form where and are uniquely determined real numbers.

In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in the 1930s by Richard Brauer and Cecil Nesbitt and were named after Ferdinand Frobenius. Tadashi Nakayama discovered the beginnings of a rich duality theory , . Jean Dieudonné used this to characterize Frobenius algebras .

Since unitals are block designs, two unitals are said to be isomorphic if there is a design isomorphism between them, that is, a bijection between the point sets which maps blocks to blocks. This concept does not take into account the property of embeddability, so to do so we say that two unitals, embedded in the same ambient plane, are equivalent if there is a collineation of the plane which maps one unital to the other.

A finite- dimensional unital associative algebra (over any field) is a division algebra if and only if it has no zero divisors. Whenever A is an associative unital algebra over the field F and S is a simple module over A, then the endomorphism ring of S is a division algebra over F; every associative division algebra over F arises in this fashion. The center of an associative division algebra D over the field K is a field containing K. The dimension of such an algebra over its center, if finite, is a perfect square: it is equal to the square of the dimension of a maximal subfield of D over the center. Given a field F, the Brauer equivalence classes of simple (contains only trivial two-sided ideals) associative division algebras whose center is F and which are finite-dimensional over F can be turned into a group, the Brauer group of the field F. One way to construct finite-dimensional associative division algebras over arbitrary fields is given by the quaternion algebras (see also quaternions).

In mathematics, the Eckmann–Hilton argument (or Eckmann–Hilton principle or Eckmann–Hilton theorem) is an argument about two unital magma structures on a set where one is a homomorphism for the other. Given this, the structures can be shown to coincide, and the resulting magma demonstrated to be commutative monoid. This can then be used to prove the commutativity of the higher homotopy groups. The principle is named after Beno Eckmann and Peter Hilton, who used it in a 1962 paper.

Let R be a ring (with unity) and let r be an element of R. Then r is said to be quasiregular, if 1 − r is a unit in R; that is, invertible under multiplication. The notions of right or left quasiregularity correspond to the situations where 1 − r has a right or left inverse, respectively. An element x of a non-unital ring is said to be right quasiregular if there is y such that x + y - xy = 0.Lam, Ex. 4.2, p.

If T is a closed operator (which includes the case that T is a bounded operator), boundedness of such inverses follows automatically if the inverse exists at all. The space of bounded linear operators B(X) on a Banach space X is an example of a unital Banach algebra. Since the definition of the spectrum does not mention any properties of B(X) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.

In abstract algebra, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring X, denoted by End(X); the set of all homomorphisms of X into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map 0: x \mapsto 0 as additive identity and the identity map 1: x \mapsto x as multiplicative identity.

A Clifford algebra is a unital associative algebra that contains and is generated by a vector space V over a field K, where V is equipped with a quadratic form . The Clifford algebra is the "freest" algebra generated by V subject to the conditionMathematicians who work with real Clifford algebras and prefer positive definite quadratic forms (especially those working in index theory) sometimes use a different choice of sign in the fundamental Clifford identity. That is, they take . One must replace Q with −Q in going from one convention to the other.

Only elements in the Green class H1 have an inverse from the unital magma perspective, whereas for any idempotent e, the elements of He have an inverse as defined in this section. Under this more general definition, inverses need not be unique (or exist) in an arbitrary semigroup or monoid. If all elements are regular, then the semigroup (or monoid) is called regular, and every element has at least one inverse. If every element has exactly one inverse as defined in this section, then the semigroup is called an inverse semigroup.

In mathematics, and more specifically in abstract algebra, a rng (or pseudo- ring or non-unital ring) is an algebraic structure satisfying the same properties as a ring, without assuming the existence of a multiplicative identity. The term "rng" (pronounced rung) is meant to suggest that it is a "ring" without "i", i.e. without the requirement for an "identity element". There is no consensus in the community as to whether the existence of a multiplicative identity must be one of the ring axioms (see the history section of the article on rings).

In mathematics, specifically in ring theory, the simple modules over a ring R are the (left or right) modules over R that are non-zero and have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M. Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory. In this article, all modules will be assumed to be right unital modules over a ring R.

In stable homotopy theory, a ring spectrum is a spectrum E together with a multiplication map :μ: E ∧ E → E and a unit map : η: S → E, where S is the sphere spectrum. These maps have to satisfy associativity and unitality conditions up to homotopy, much in the same way as the multiplication of a ring is associative and unital. That is, : μ (id ∧ μ) ∼ μ (μ ∧ id) and : μ (id ∧ η) ∼ id ∼ μ(η ∧ id). Examples of ring spectra include singular homology with coefficients in a ring, complex cobordism, K-theory, and Morava K-theory.

Let X be a set equipped with two binary operations, which we will write \circ and \otimes, and suppose: # \circ and \otimes are both unital, meaning that there are elements 1_\circ and 1_\otimes of X such that 1_\circ \circ a= a =a \circ 1_\circ and 1_\otimes \otimes a= a =a \otimes 1_\otimes, for all a\in X. # (a \otimes b) \circ (c \otimes d) = (a \circ c) \otimes (b \circ d) for all a,b,c,d \in X . Then \circ and \otimes are the same and in fact commutative and associative.

In mathematics, the peak algebra is a (non-unital) subalgebra of the group algebra of the symmetric group Sn, studied by . It consists of the elements of the group algebra of the symmetric group whose coefficients are the same for permutations with the same peaks. (Here a peak of a permutation σ on {1,2,...,n} is an index i such that σ(i–1)<σ(i)>σ(i+1).) It is a left ideal of the descent algebra. The direct sum of the peak algebras for all n has a natural structure of a Hopf algebra.

Also, one may consider classical groups over a unital associative algebra R over F; where R = H (an algebra over reals) represents an important case. For the sake of generality the article will refer to groups over R, where R may be the ground field F itself. Considering their abstract group theory, many linear groups have a "special" subgroup, usually consisting of the elements of determinant 1 over the ground field, and most of them have associated "projective" quotients, which are the quotients by the center of the group. For orthogonal groups in characteristic 2 "S" has a different meaning.

If G is a finite group and k a field with characteristic 0, then one shows in the theory of group representations that any subrepresentation of a given one is already a direct summand of the given one. Translated into module language, this means that all modules over the group algebra kG are injective. If the characteristic of k is not zero, the following example may help. If A is a unital associative algebra over the field k with finite dimension over k, then Homk(−, k) is a duality between finitely generated left A-modules and finitely generated right A-modules.

For example, # Banach's extension theorem which is used to prove one of the most fundamental results in functional analysis, the Hahn–Banach theorem # Every vector space has a basis, a result from linear algebra (to which it is equivalent). In particular, the real numbers, as a vector space over the rational numbers, possess a Hamel basis. # Every commutative unital ring has a maximal ideal, a result from ring theory # Tychonoff's theorem in topology (to which it is also equivalent) # Every proper filter is contained in an ultrafilter, a result that yields completeness theorem of first-order logicJ.L. Bell & A.B. Slomson (1969).

The Gaussian elimination algorithm remains applicable. The column rank of a matrix is the dimension of the right module generated by the columns, and the row rank is dimension of the left module generated by the rows; the same proof as for the vector space case can be used to show that these ranks are the same, and define the rank of a matrix. In fact the converse is also true and this gives a characterization of division rings via their module category: A unital ring R is a division ring if and only if every R-module is free.Grillet, Pierre Antoine.

For example, the general linear group over R (the set of real numbers) is the group of invertible matrices of real numbers, and is denoted by GLn(R) or . More generally, the general linear group of degree n over any field F (such as the complex numbers), or a ring R (such as the ring of integers), is the set of invertible matrices with entries from F (or R), again with matrix multiplication as the group operation.Here rings are assumed to be associative and unital. Typical notation is GLn(F) or , or simply GL(n) if the field is understood.

Every simple ring R with unity is both left and right primitive. (However, a simple non-unital ring may not be primitive.) This follows from the fact that R has a maximal left ideal M, and the fact that the quotient module R/M is a simple left R-module, and that its annihilator is a proper two-sided ideal in R. Since R is a simple ring, this annihilator is {0} and therefore R/M is a faithful left R-module. Weyl algebras over fields with characteristic zero are primitive, and since they are domains, they are examples without minimal one- sided ideals.

In mathematics, a mutation, also called a homotope, of a unital Jordan algebra is a new Jordan algebra defined by a given element of the Jordan algebra. The mutation has a unit if and only if the given element is invertible, in which case the mutation is called a proper mutation or an isotope. Mutations were first introduced by Max Koecher in his Jordan algebraic approach to Hermitian symmetric spaces and bounded symmetric domains of tube type. Their functorial properties allow an explicit construction of the corresponding Hermitian symmetric space of compact type as a compactification of a finite-dimensional complex semisimple Jordan algebra.

If A is unital, then ∂(1) = 0 since ∂(1) = ∂(1 × 1) = ∂(1) + ∂(1). For example, in a differential field of characteristic zero K, the rationals are always a subfield of the field of constants of K. Any ring is a differential ring with respect to the trivial derivation which maps any ring element to zero. The field Q(t) has a unique structure as a differential field, determined by setting ∂(t) = 1: the field axioms along with the axioms for derivations ensure that the derivation is differentiation with respect to t. For example, by commutativity of multiplication and the Leibniz law one has that ∂(u2) = u ∂(u) + ∂(u)u = 2u∂(u).

For instance, the ring (in fact field) of complex numbers, which can be constructed from the polynomial ring over the real numbers by factoring out the ideal of multiples of the polynomial . Another example is the construction of finite fields, which proceeds similarly, starting out with the field of integers modulo some prime number as the coefficient ring (see modular arithmetic). If is commutative, then one can associate with every polynomial in a polynomial function with domain and range equal to . (More generally, one can take domain and range to be any same unital associative algebra over .) One obtains the value by substitution of the value for the symbol in .

We recall that for a (unital, associative) ring R, we denote by V(R) the (conical, commutative) monoid of isomorphism classes of finitely generated projective right R-modules, see here for more details. Recall that if R is von Neumann regular, then V(R) is a refinement monoid. Denote by Idc R the (∨,0)-semilattice of finitely generated two-sided ideals of R. We denote by L(R) the lattice of all principal right ideals of a von Neumann regular ring R. It is well known that L(R) is a complemented modular lattice. The following result was observed by Wehrung, building on earlier works mainly by Jónsson and Goodearl.

Let N be a rational prime, and define :J0(N) = J as the Jacobian variety of the modular curve :X0(N) = X. There are endomorphisms Tl of J for each prime number l not dividing N. These come from the Hecke operator, considered first as an algebraic correspondence on X, and from there as acting on divisor classes, which gives the action on J. There is also a Fricke involution w (and Atkin–Lehner involutions if N is composite). The Eisenstein ideal, in the (unital) subring of End(J) generated as a ring by the Tl, is generated as an ideal by the elements : Tl − l - 1 for all l not dividing N, and by :w + 1.

In ring theory and Frobenius algebra extensions, areas of mathematics, there is a notion of depth two subring or depth of a Frobenius extension. The notion of depth two is important in a certain noncommutative Galois theory, which generates Hopf algebroids in place of the more classical Galois groups, whereas the notion of depth greater than two measures the defect, or distance, from being depth two in a tower of iterated endomorphism rings above the subring. A more recent definition of depth of any unital subring in any associative ring is proposed (see below) in a paper studying the depth of a subgroup of a finite group as group algebras over a commutative ring.

A finite-dimensional, unital, associative algebra A defined over a field k is said to be a Frobenius algebra if A is equipped with a nondegenerate bilinear form σ:A × A → k that satisfies the following equation: σ(a·b,c)=σ(a,b·c). This bilinear form is called the Frobenius form of the algebra. Equivalently, one may equip A with a linear functional λ : A → k such that the kernel of λ contains no nonzero left ideal of A. A Frobenius algebra is called symmetric if σ is symmetric, or equivalently λ satisfies λ(a·b) = λ(b·a). There is also a different, mostly unrelated notion of the symmetric algebra of a vector space.

The definition is equivalent to saying that a unital associative R-algebra is a monoid object in R-Mod (the monoidal category of R-modules). By definition, a ring is a monoid object in the category of abelian groups; thus, the notion of an associative algebra is obtained by replacing the category of abelian groups with the category of modules. Pushing this idea further, some authors have introduced a "generalized ring" as a monoid object in some other category that behaves like the category of modules. Indeed, this reinterpretation allows one to avoid making an explicit reference to elements of an algebra A. For example, the associativity can be expressed as follows.

In mathematics, a composition ring, introduced in , is a commutative ring (R, 0, +, −, ·), possibly without an identity 1 (see non-unital ring), together with an operation : \circ: R \times R \rightarrow R such that, for any three elements f,g,h\in R one has # (f+g)\circ h=(f\circ h)+(g\circ h) # (f\cdot g)\circ h = (f\circ h)\cdot (g\circ h) # (f\circ g)\circ h = f\circ (g\circ h). It is not generally the case that f\circ g=g\circ f, nor is it generally the case that f\circ (g+h) (or f\circ (g\cdot h)) has any algebraic relationship to f\circ g and f\circ h.

A year later, he was given political appointments in two government agencies, in 1967, he was managing director of WEMABOD, a regional property and investment company and also in 1967, he succeeded Kola Balogun as chairman of Nigerian National Shipping Line. After leaving WEMABOD, he became an investor in various firms including AGIP petroleum marketing and NCR Nigeria. He also founded the private firms Nigerlink Industries, Unital Builders and a holding company Lagos Investments. After the Nigerian Enterprise Promotion Act, he took equity interest in some foreign companies operating in Nigeria such as investments in the Nigerian operations of Bowring Group, Inchape, Schlumberger, Phoenix Assurance, UTC Nigeria, Evans Brothers and Seven-Up.

In mathematics, an H-space,The H in H-space was suggested by Jean-Pierre Serre in recognition of the influence exerted on the subject by Heinz Hopf (see J. R. Hubbuck. "A Short History of H-spaces", History of topology, 1999, pages 747–755). or a topological unital magma, is a topological space X (generally assumed to be connected) together with a continuous map μ : X × X → X with an identity element e such that μ(e, x) = μ(x, e) = x for all x in X. Alternatively, the maps μ(e, x) and μ(x, e) are sometimes only required to be homotopic to the identity (in this case e is called homotopy identity), sometimes through basepoint preserving maps. These three definitions are in fact equivalent for H-spaces that are CW complexes.

A ring homomorphism (of unital, but not necessarily commutative rings) :K \to A is called separable (or a separable extension) if the multiplication map :\mu : A \otimes_K A \to A, a \otimes b \mapsto ab admits a section :\sigma: A \to A \otimes_K A by means of a homomorphism σ of A-A-bimodules. Such a section σ is determined by its value :p := \sigma(1) = \sum a_i \otimes b_i σ(1). The condition that σ is a section of μ is equivalent to :\sum a_i b_i = 1 and the condition to be an homomorphism of A-A-bimodules is equivalent to the following requirement for any a in A: :\sum a a_i \otimes b_i = \sum a_i \otimes b_i a. Such an element p is called a separability idempotent, since it satisfies p^2 = p.

Commutative algebra (in the form of polynomial rings and their quotients, used in the definition of algebraic varieties) has always been a part of algebraic geometry. However, in the late 1950s, algebraic varieties were subsumed into Alexander Grothendieck's concept of a scheme. Their local objects are affine schemes or prime spectra, which are locally ringed spaces, which form a category that is antiequivalent (dual) to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a field k, and the category of finitely generated reduced k-algebras. The gluing is along the Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes.

In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations. Hopf algebras occur naturally in algebraic topology, where they originated and are related to the H-space concept, in group scheme theory, in group theory (via the concept of a group ring), and in numerous other places, making them probably the most familiar type of bialgebra. Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems on the other.

Their local objects are affine schemes or prime spectra which are locally ringed spaces which form a category which is antiequivalent to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a field k, and the category of finitely generated reduced k-algebras. The gluing is along Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes. The Zariski topology in the set theoretic sense is then replaced by a Grothendieck topology. Grothendieck introduced Grothendieck topologies having in mind more exotic but geometrically finer and more sensitive examples than the crude Zariski topology, namely the étale topology, and the two flat Grothendieck topologies: fppf and fpqc; nowadays some other examples became prominent including Nisnevich topology.

In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field. The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over the field K. A standard first example of a K-algebra is a ring of square matrices over a field K, with the usual matrix multiplication. A commutative algebra is an associative algebra that has a commutative multiplication, or, equivalently, an associative algebra that is also a commutative ring. In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification.

A version of the lemma holds for right modules over non-commutative unital rings R. The resulting theorem is sometimes known as the Jacobson–Azumaya theorem. Let J(R) be the Jacobson radical of R. If U is a right module over a ring, R, and I is a right ideal in R, then define U·I to be the set of all (finite) sums of elements of the form u·i, where · is simply the action of R on U. Necessarily, U·I is a submodule of U. If V is a maximal submodule of U, then U/V is simple. So U·J(R) is necessarily a subset of V, by the definition of J(R) and the fact that U/V is simple. Thus, if U contains at least one (proper) maximal submodule, U·J(R) is a proper submodule of U. However, this need not hold for arbitrary modules U over R, for U need not contain any maximal submodules.

Let C be the category of vector spaces K-Vect over a field K and let D be the category of algebras K-Alg over K (assumed to be unital and associative). Let :U : K-Alg -> K-Vect be the forgetful functor which assigns to each algebra its underlying vector space. Given any vector space V over K we can construct the tensor algebra T(V). The tensor algebra is characterized by the fact: :“Any linear map from V to an algebra A can be uniquely extended to an algebra homomorphism from T(V) to A.” This statement is an initial property of the tensor algebra since it expresses the fact that the pair (T(V),i), where i:V \to U(T(V)) is the inclusion map, is a universal morphism from the vector space V to the functor U. Since this construction works for any vector space V, we conclude that T is a functor from K-Vect to K-Alg.

A subset S of R is said to be a subring if it can be regarded as a ring with the addition and the multiplication restricted from R to S. Equivalently, S is a subring if it is not empty, and for any x, y in S, xy, x+y and -x are in S. If all rings have been assumed, by convention, to have a multiplicative identity, then to be a subring one would also require S to share the same identity element as R.In the unital case, like addition and multiplication, the multiplicative identity must be restricted from the original ring. The definition is also equivalent to requiring the set- theoretic inclusion is a ring homomorphism. So if all rings have been assumed to have a multiplicative identity, then a proper ideal is not a subring. For example, the ring Z of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Z[X] (in both cases, Z contains 1, which is the multiplicative identity of the larger rings).

A unital subring B \subseteq A has (or is) right depth two if there is a split epimorphism of natural A-B-bimodules from A^n \rightarrow A \otimes_B A for some positive integer n; by switching to natural B-A-bimodules, there is a corresponding definition of left depth two. Here we use the usual notation A^n = A \times \ldots \times A (n times) as well as the common notion, p is a split epimorphism if there is a homomorphism q in the reverse direction such that pq = identity on the image of p. (Sometimes the subring B in A is referred to as the ring extension A over B; the theory works as well for a ring homomorphism B into A, which induces right and left B-modules structures on A.) Equivalently, the condition for left or right depth two may be given in terms of a split monomorphism of bimodules where the domains and codomains above are reversed. For example, let A be the group algebra of a finite group G (over any commutative base ring k; see the articles on group theory and group ring for the elementary definitions).

In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals. In other words, I is a maximal ideal of a ring R if there are no other ideals contained between I and R. Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields. In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one sided maximal ideal A is not necessarily two-sided, the quotient R/A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J(R).

Any semisimple algebra over the complex numbers C of finite dimension can be expressed as a direct sum ⊕k Mnk(C) of matrix algebras, and the C-algebra homomorphisms between two such algebras up to inner automorphisms on both sides are completely determined by the multiplicity number between 'matrix algebra' components. Thus, an injective homomorphism of ⊕k=1i Mnk(C) into ⊕l=1j Mml(C) may be represented by a collection of positive numbers ak, l satisfying ∑ nk ak, l ≤ ml. (The equality holds if and only if the homomorphism is unital; we can allow non-injective homomorphisms by allowing some ak,l to be zero.) This can be illustrated as a bipartite graph having the vertices marked by numbers (nk)k on one hand and the ones marked by (ml)l on the other hand, and having ak, l edges between the vertex nk and the vertex ml. Thus, when we have a sequence of finite- dimensional semisimple algebras An and injective homomorphisms φn : An' → An+1: between them, we obtain a Bratteli diagram by putting : Vn = the set of simple components of An (each isomorphic to a matrix algebra), marked by the size of matrices.