Lesson plan
Singular submodule
In the branches of abstract algebra known as ring theory and module theory, each right (resp. left) 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, \operatorname{tors}(M) = \mathcal{Z}(M) . If R is any ring, \mathcal{Z}(R_R) is defined considering R as a right module, and in this case \mathcal{Z}(R_R) is a two-sided ideal of R called the right singular ideal of R. The left handed analogue \mathcal{Z}(_R R) is defined similarly. It is possible for \mathcal{Z}(R_R) \neq \mathcal{Z}(_R R) .
Article summary
In the branches of abstract algebra known as ring theory and module theory, each right (resp. left) 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, \operatorname{tors}(M) = \mathcal{Z}(M) . If R is any ring, \mathcal{Z}(R_R) is defined considering R as a right module, and in this case \mathcal{Z}(R_R) is a two-sided ideal of R called the right singular ideal of R. The left handed analogue \mathcal{Z}(_R R) is defined similarly. It is possible for \mathcal{Z}(R_R) \neq \mathcal{Z}(_R R) .
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Discussion questions
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