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...

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

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

branches of mathematics such as geometry and analysis. Rings appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains...

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...

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...

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

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

commutative rings that do not extend to noncommutative rings. Commutative rings appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings...

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

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...

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

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...

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...

for Noetherian rings, and has proved to be a powerful generalization of the notion of direct product.[citation needed] A subdirect product is a subalgebra...

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...

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

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

category of rings, characterizes the ring  Z {\displaystyle \mathbb {Z} } . This unique homomorphism is injective if and only if the characteristic of the...

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...

constructed in a similar fashion. The coordinate rings of algebraic varieties are important examples of quotient rings in algebraic geometry. As a simple case...

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...

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...

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

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

described for the rings of integers of number fields. In fact, class field theory is the study of such groups of class rings. For the ring of integerspg 2...

In 2017, ADT sued Ring for trade secret theft in the case of ADT vs. Mike Harris relating to Ring Alarm. Sales of the alarm product were preliminarily...

Cartesian product of the underlying sets, equipped with a structure of the same type. For example, direct product of rings, direct product of topological...