Group ring - Wikipedia
WEBIn 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, its ring of scalars is the given ring, and its basis is the set of elements of the given group.
WEB1. The group ring k[G] The main idea is that representations of a group G over a field k are “the same” as modules over the group ring k[G]. First I defined both terms. 1.1. Representations of groups. Definition 1.1. A representation of a group G over a field k is defined to be a group homomorphism : G Autk(V ) where V is a vector space over k.
Group Ring -- from Wolfram MathWorld
WEBAug 22, 2024 · Group Ring. The set of sums ranging over a multiplicative group and are elements of a field with all but a finite number of . Group rings are graded algebras.
group ring - PlanetMath.org
WEBFeb 9, 2018 · group ring. For any group G G, the group ring Z[G] ℤ [G] is defined to be the ring whose additive group is the abelian group of formal integer linear combinations of elements of G G, and whose multiplication operation is defined by multiplication in G G, extended Z ℤ –linearly to Z[G] ℤ [G].
What are the differences between rings, groups, and fields?
WEBIn fact, every ring is a group, and every field is a ring. A ring is an abelian group with an additional operation, where the second operation is associative and the distributive property make the two operations "compatible".
WEBA RING is a set equipped with two operations, called addition and multiplication. A RING is a GROUP under addition and satisfies some of the properties of a group for multiplication. A FIELD is a GROUP under both addition and multiplication. Definition 1. A GROUP is a set G which is CLOSED under an operation ∗ (that is, for
WEBIf G is a group, the R-module R[G] is a ring with r1g1 r2g2:= (r1r2)(g1g2) (to define the product of elements or R [ G ] use the distributive law), called the group ring R [ G ] of G over R ; note that this ring is commutative if and only if G is.
WEBtheory to the ring-theoretic structure of group rings. The problems are fairly old and may now be solvable given the present state of the subject. A general reference for character theory is of course Marty’s book [6], while [10] affords a general reference for group rings. 1. Character Regular p-Groups
Group Algebra -- from Wolfram MathWorld
WEBAug 22, 2024 · The group algebra K[G], where K is a field and G a group with the operation *, is the set of all linear combinations of finitely many elements of G with coefficients in K, hence of all elements of the form a_1g_1+a_2g_2+...+a_ng_n, (1) where a_i in K and g_i in G for all i=1,...,n.
Group Rings, An Introduction - mathreference.com
WEBThe group ring of a ring R and a group G, denoted RG, consists of formal linear combinations of elements of G with coefficients taken from R. Addition occurs per component, i.e. gather terms together according to the elements of G, and add up the coefficients in R.
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