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In 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 one-to-one with the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring. If the given ring is commutative, a group ring is also referred to as a group algebra, for it is indeed an algebra over the given ring. The apparatus of group rings is especially useful in the theory of group representations. ==Definition== Let ''G'' be a group, written multiplicatively, and let ''R'' be a ring. The group ring of ''G'' over ''R'', which we will denote by ''R''() (or simply RG), is the set of mappings ''f'' : ''G'' → ''R'' of finite support,〔Polcino & Sehgal (2002), p. 131.〕 where the product α''f'' of a scalar α in ''R'' and a vector (or mapping) ''f'' is defined as the vector , and the sum of two vectors ''f'' and ''g'' is defined as the vector . To turn the additive group ''R''() into a ring, we define the product of ''f'' and ''g'' to be the vector : The summation is legitimate because ''f'' and ''g'' are of finite support, and the ring axioms are readily verified. Some variations in the notation and terminology are in use. In particular, the mappings such as ''f'' : ''G'' → ''R'' are sometimes written as what are called "formal linear combinations of elements of ''G'', with coefficients in ''R''":〔Polcino & Sehgal (2002), p. 129 and 131.〕 : or simply : where this doesn't cause confusion.〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Group ring」の詳細全文を読む スポンサード リンク
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