In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

Let R be a fixed commutative ring.

If A itself is commutative ( as a ring ) then it is called a commutative R-algebra.

* Any commutative ring R is an algebra over itself, or any subring of R.

* Any ring of matrices with coefficients in a commutative ring R forms an R-algebra under matrix addition and multiplication.

* Every polynomial ring R ..., x < sub > n </ sub > is a commutative R-algebra.

The definition works without any changes if instead of vector spaces over a field F, we use modules over a commutative ring R. It also can be easily generalized to n-ary functions, where the proper term is multilinear.

Let A be a unital commutative Banach algebra over C. Since A is then a commutative ring with unit, every non-invertible element of A belongs to some maximal ideal of A.

( valid for any elements x, y of a commutative ring ),

* The spectrum of any commutative ring with the Zariski topology ( that is, the set of all prime ideals ) is compact, but never Hausdorff ( except in trivial cases ).

A ring homomorphism of commutative rings determines a morphism of Kähler differentials which sends an element dr to d ( f ( r )), the exterior differential of f ( r ).

Two ideals A and B in the commutative ring R are called coprime ( or comaximal ) if A + B = R. This generalizes Bézout's identity: with this definition, two principal ideals ( a ) and ( b ) in the ring of integers Z are coprime if and only if a and b are coprime.

Although most often used for matrices whose entries are real or complex numbers, the definition of the determinant only involves addition, subtraction and multiplication, and so it can be defined for square matrices with entries taken from any commutative ring.

For square matrices with entries in a non-commutative ring, for instance the quaternions, there is no unique definition for the determinant, and no definition that has all the usual properties of determinants over commutative rings.

Provided the underlying scalars form a field ( more generally, a commutative ring with unity ), the definition below shows that such a function exists, and it can be shown to be unique.

The center of a division ring is commutative and therefore a field.

In abstract algebra, a field is a commutative ring which contains a multiplicative inverse for every nonzero element, equivalently a ring whose nonzero elements form an abelian group under multiplication.

The notion of greatest common divisor can more generally be defined for elements of an arbitrary commutative ring, although in general there need not exist one for every pair of elements.

For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing ( the more general ) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.

* Every commutative real unital Noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers.

Relations among morphisms ( such as ) are often depicted using commutative diagrams, with " points " ( corners ) representing objects and " arrows " representing morphisms.

The latter case with the function f can be expressed by a commutative triangle.

Intuitively, a field is a set F that is a commutative group with respect to two compatible operations, addition and multiplication, with " compatible " being formalized by distributivity, and the caveat that the additive identity ( 0 ) has no multiplicative inverse ( one cannot divide by 0 ).

This extension of the definition is also compatible with the generalization for commutative rings given below.

* Let A be a commutative ring with unity and let S be a multiplicative subset of A.

* The ring of formal power series over a commutative ring R can be thought of as the inverse limit of the rings, indexed by the natural numbers as usually ordered, with the morphisms from to given by the natural projection.

This applies to the I-indexed diagrams in the category of R-modules, with R a commutative ring ; it is not necessarily true in an arbitrary abelian category ( see Roos ' " Derived functors of inverse limits revisited " for examples of abelian categories in which lim ^ n, on diagrams indexed by a countable set, is nonzero for n > 1 ).

Any commutative monoid is endowed with its algebraic preordering ≤, defined by x ≤ y if and only if there exists z such that x + z = y.

Thus the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R. The map from R to R sending r to rX < sup > 0 </ sup > is an injective homomorphism of rings, by which R is viewed as a subring of R. If R is commutative, then R is an algebra over R.

If R is commutative, then one can associate to every polynomial P in R, a polynomial function f with domain and range equal to R ( more generally one can take domain and range to be the same unital associative algebra over R ).

A simple with the individual keys is such a commutative cipher.

A multiplicative identity is not required for the role of the integral domain ; this construction can be applied to any non-trivial commutative pseudo-ring with no zero divisors.

* An internal operation ( addition ) which is associative, commutative, distributive and with zero and unity elements

and commutative ring with unity.

The construction of the integral cycle space can be carried out for any field, abelian group, or ( most generally ) commutative ring ( with unity ) R replacing the integers.

Let A be a commutative Noetherian ring with unity.

A near-ring is a ring ( not necessarily with unity ) if and only if addition is commutative and multiplication is distributive over addition on the left.

* In the case of commutative rings with unity, this example is prototypical in the following sense: the Chinese remainder theorem shows that for a semi-local commutative ring R with unit and maximal ideals m < sub > 1 </ sub >, ..., m < sub > n </ sub >

A module over a ( not necessarily commutative ) ring with unity is said to be semisimple ( or completely reducible ) if it is the direct sum of simple ( irreducible ) submodules.

Let R be a commutative ring with unity, and let M, N and L be three R-modules.

Let A and B be two commutative rings with unity, and let f: A → B be a ( unital ) ring homomorphism.