Lexic.us

Definition of Group ring

1. Noun. (algebra) Given ring ''R'' with identity not equal to zero, and group $G\; =\; \backslash \{g\_1,\; g\_2,\; ...,\; g\_n\backslash \}$, the group ring ''RG'' has elements of the form $a\_1\; g\_1\; +\; a\_2\; g\_2\; +\; ...\; +\; a\_n\; g\_n$ (where $a\_i\; \backslash isin\; R$) such that the sum of $a\_1\; g\_1\; +\; a\_2\; g\_2\; +\; ...\; +\; a\_n\; g\_n$ and $b\_1\; g\_1\; +\; b\_2\; g\_2\; +\; ...\; +\; b\_n\; g\_n$ is $(a\_1\; +\; b\_1)\; g\_1\; +\; (a\_2\; +\; b\_2)\; g\_2\; +\; ...\; +\; (a\_n\; +\; b\_n)\; g\_n$ and the product is $\backslash sum\_\{k=1\}^n\; \backslash left\; (\; \backslash sum\_\{g\_i\; g\_j\; =\; g\_k\}\; a\_i\; b\_j\; \backslash right\; )\; g\_k$. ¹

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