Singular submodule

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 $\mathcal{Z}(M)=\{m\in M \mid \mathrm{ann}(m)\subseteq_e R\}\,$ . For general rings, $\mathcal{Z}(M)$ 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, $tors(M)=\mathcal{Z}(M)$ .

单词	Singular submodule
释义	Singular submodule 英语百科 Singular submodule 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 $\mathcal{Z}(M)=\{m\in M \mid \mathrm{ann}(m)\subseteq_e R\}\,$ . For general rings, $\mathcal{Z}(M)$ 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, $tors(M)=\mathcal{Z}(M)$ .
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