Left regular module
= Left regular module
{title2=${}_AA$}
The left regular <module> of a unital <associative algebra> $A$ is its underlying additive group with $A$ acting by left multiplication. Its <submodules> are exactly the <left ideals> of $A$. The <associative algebra> is a <semisimple algebra> precisely when this <module> is a <semisimple module>.