Nonsingular ring

In the branches of abstract algebra known as ring theory and module theory, each right (resp. left) R module M has a singular submodule consisting of elements whose annihilators are essential right (resp. left) ideals in R. In set notation it is usually denoted as ${\displaystyle {\mathcal {Z}}(M)=\{m\in M\mid \mathrm {ann} (m)\subseteq _{e}R\}\,}$. For general rings, ${\displaystyle {\mathcal {Z}}(M)}$ is a good generalization of the torsion submodule tors(M) which is most often defined for domains. In the case that R is a commutative domain, ${\displaystyle tors(M)={\mathcal {Z}}(M)}$.

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