mathematics, a product of rings or direct product of rings is a ring that is formed by the Cartesian product of the underlying sets of several rings (possibly...

product (or knit product), a generalization of the direct and semidirect products Product of rings Ideal operations, for product of ideals Scalar multiplication...

underlying rings are graded-commutative rings, the tensor product A ⊗ R B {\displaystyle A\otimes _{R}B} becomes a graded commutative ring by defining...

ideas of algebraic number theory and algebraic geometry. The simplest commutative rings are those that admit division by non-zero elements; such rings are...

the colimit of a span and the pullback is the limit of a cospan, we can think of the tensor product of rings and the fibered product of rings (see the examples...

mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that...

semidirect product, knit product and wreath product the free product of groups the product of rings the product of ideals the product of topological...

beyond number rings to the setting of polynomial rings and other commutative rings by David Hilbert and especially Emmy Noether. Given a ring R, a left ideal...

nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide...

number of fundamental properties of commutative rings that do not extend to noncommutative rings. A ring is a set R {\displaystyle R} equipped with two...

regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, have been introduced for generalizing some properties of polynomial...

non-associative Z {\displaystyle \mathbb {Z} } -algebras as non-associative rings. Ring-like structures with two binary operations and no other restrictions...

artinian semiprimitive ring is just a semisimple ring. Semiprimitive rings can be understood as subdirect products of primitive rings, which are described...

"ordinary" integers, the prototypical object for all those rings. It is a consequence of the ambiguity of the word "integer" in abstract algebra. Alaca & Williams...

need only be a ring, so the module concept represents a significant generalization. In commutative algebra, both ideals and quotient rings are modules,...

(Quotient ring notation always uses a fraction slash "/".) Quotient rings are distinct from the so-called "quotient field", or field of fractions, of an integral...

noncommutative rings, including the properties that apply also to commutative rings. Sometimes the term noncommutative ring is used instead of ring to refer...

Fundamental structures of algebra and discrete mathematics. Wiley. p. 128. ISBN 0-471-57180-6. Grillet, Pierre Antoine (2007). "3.5 Rings: Polynomials in One...

ideal of R. Thus, it makes sense to speak of the quotient ring R / (ker f). The first isomorphism theorem for rings states that this quotient ring is naturally...

k. Then A is an Artinian ring. As A is Artinian, if it is commutative, then it is a finite product of Artinian local rings whose residue fields are algebras...

to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third...

mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is...

category of rings, and should not be written as a direct sum. (The coproduct in the category of commutative rings is the tensor product of rings. In the...

integers. Ring theory studies the structure of rings; their representations, or, in different language, modules; special classes of rings (group rings, division...

In the category of rings, the zero ring is the terminal object, whereas the ring of integers Z is the initial object. The zero ring, denoted {0} or simply...

forgetful functor from the category of R-algebras to the category of sets. Free algebras over division rings are free ideal rings. Cofree coalgebra Tensor algebra...

In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the...

in ring theory, the crossed product of rings. This is constructed in the natural way from the group ring for a semidirect product of groups. The ring-theoretic...

of Y {\displaystyle Y} . This asymmetry disappears if the power series ring in Y {\displaystyle Y} is given the product topology where each copy of Z...

define rings without requiring the existence of a multiplicative identity (see Ring (mathematics) § History). In this case, a subring of R is a subset of R...