Editing Group ring (section)

==Definition==
Let <math>G</math> be a group, written multiplicatively, and let <math>R</math> be a ring. The group ring of <math>G</math> over <math>R</math>, which we will denote by <math>R[G]</math>, or simply <math>RG</math>, is the set of mappings <math>f\colon G \to R</math> of [[Support (mathematics)#Generalization|finite support]] (<math>f(g)</math> is nonzero for only finitely many elements <math>g</math>), where the module scalar product <math>\alpha f </math> of a scalar <math>\alpha</math> in <math>R</math> and a mapping <math>f</math> is defined as the mapping <math>x \mapsto \alpha \cdot f(x)</math>, and the module group sum of two mappings <math>f</math> and <math>g</math> is defined as the mapping <math>x \mapsto f(x) + g(x)</math>. To turn the additive group <math>R[G]</math> into a ring, we define the product of <math>f</math> and <math>g</math> to be the mapping
:<math>x\mapsto\sum_{uv=x}f(u)g(v)=\sum_{u\in G}f(u)g(u^{-1}x).</math>
The summation is legitimate because <math>f</math> and <math>g</math> 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 <math>f : G \to R</math> are sometimes<ref>Milies & Sehgal (2002), pp. 129 and 131.</ref> written as what are called "formal linear combinations of elements of <math>G</math> with coefficients in <math>R</math>
":
:<math>\sum_{g\in G}f(g) g,</math>
or simply
:<math>\sum_{g\in G}f_g g.</math><ref name=Milies>Milies & Sehgal (2002), p. 131.</ref>

Note that if the ring <math>R</math> is in fact a field <math>K</math>, then the module structure of the group ring <math>RG</math> is in fact a vector space over <math>K</math>.