Left ideal

Summary

A left ideal of a ring $R$ is an additive subgroup that is closed under multiplication on the left by every element of $R$. Left ideals are exactly the submodules of $R$ viewed as a left module over itself. In a commutative ring left ideals, right ideals and two-sided ideals all coincide with ideals. In a noncommutative ring the three notions differ, and left ideals carry much of the module theory of $R$.