Let G be a group and R a nontrivial commutative ring with un

Let G be a group and R a non-trivial commutative ring with unit. We will always write the peration of G as multiplication. Let RG = {r191 + r292 + .. . rkIk | k E Z20, ri E R, and gi E G} We view rigi +gh simply as a formal sum, ie. it is a multivariable poly nomial with coefficients in R and variables in G. We can then define an addition on RG componentuise, .e. (One should note that a priori two elements of RG will not contain the same variables. When adding we may assume that the same variables appear in both elements by just adding ORg when necessary.) Multiplication is defined polynomially, i.e. risj9k Note that the second sum has only finitely many terms with non-zero coefficient

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