Regular module

The regular left module of a ring $R$ is the left $R$-module whose underlying group is the additive abelian group $R$, with multiplication given by left multiplication from $R$. The right regular module is defined similarly. The left regular $R$-module is sometimes denoted ${_R R}$, and the right regular $R$-module is sometimes denoted $R_R$. If $R$ is a commutative ring, then the two structures are the same structure, called simply the regular $R$-module.

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