algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module...

A ring may be defined as a set that is endowed with two binary operations called addition and multiplication such that the ring is an abelian group with...

n-dimensional formal group law gives an n-dimensional Lie algebra over the ring R, defined in terms of the quadratic part F2 of the formal group law. [x,y] =...

or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists...

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

commutative) ring. The concept of a module also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers...

a monoid ring is a ring constructed from a ring and a monoid, just as a group ring is constructed from a ring and a group. Let R be a ring and let G...

homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and other algebraic structures...

In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied...

particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R i {\displaystyle R_{i}} such...

Ring Ring is the debut studio album by the Swedish group ABBA, initially credited as Björn & Benny, Agnetha & Frida. It was released in Scandinavia on...

algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series...

congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense...

ring to extensions over another, especially the group ring of a group and of a subgroup. It thus relates the group cohomology with respect to a group...

A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as group rings of finite groups over...

the group ring to a completion of the exp ring. However in general the Exp ring can be much larger than the group ring: for example, the group ring of...

specific operations acting on their elements. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field...

category generalizes the opposite group, opposite ring, etc. In this section the symbol for multiplication in the opposite ring is changed from asterisk to...

representations of the group. As such, they are similar to the group ring associated to a discrete group. If G is a locally compact Hausdorff group, G carries an...

mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring of X, denoted by End(X); the set of all homomorphisms...

{\displaystyle \mathbb {Z} [G]} , for any group G {\displaystyle G} , is a ring whose additive group is the free abelian group over G {\displaystyle G} . When G...

distinguishes between different types of algebraic structures, such as groups, rings, and fields, based on the number of operations they use and the laws...

United States used the purity ring as a symbol of commitment. In particular, Catholic and evangelical Christian groups which promoted virginity pledges...

The Lord of the Rings: The Fellowship of the Ring is a 2001 epic high fantasy adventure film directed by Peter Jackson from a screenplay by Fran Walsh...

closely related to group cohomology: in Quillen's +-construction of K-theory, K-theory of a ring R is defined as the homotopy groups of a space B G L (...

In mathematics, an order in the sense of ring theory is a subring O {\displaystyle {\mathcal {O}}} of a ring A {\displaystyle A} , such that A {\displaystyle...

congruent to 1 modulo 4 is isomorphic to the group ring (Z/2Z)[V] where V is the Klein 4-group. The Witt ring of a local field with maximal ideal of norm...

as representation theory, the representation ring (or Green ring after J. A. Green) of a group is a ring formed from all the (isomorphism classes of the)...

abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally...

noncommutative rings: The matrix ring of n-by-n matrices over the real numbers, where n > 1 Hamilton's quaternions Any group ring constructed from a group that...