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In the branches of abstract algebra known as ring theory and module theory, each right R-module M has a singular submodule consisting of elements whose annihilators are essential right ideals in R. In set notation it is usually denoted as Z = { m ∈ M ∣ a n n ⊆ e R } {\displaystyle {\mathcal {Z}}=\{m\in M\mid \mathrm {ann} \subseteq _{e}R\}\,}. For general rings, Z {\displaystyle {\mathcal {Z}}} is a good generalization of the torsion submodule tors which is most often defined for domains. In the case that R is a commutative domain, tors = Z {\displaystyle \operatorname {tors} ={\mathcal {Z}}}.
If R is any ring, Z {\displaystyle {\mathcal {Z}}} is defined considering R as a right module, and in this case Z {\displaystyle {\mathcal {Z}}} is a two-sided ideal of R called the right singular ideal of R. The left handed analogue Z {\displaystyle {\mathcal {Z}}} is defined similarly. It is possible for Z ≠ Z {\displaystyle {\mathcal {Z}}\neq {\mathcal {Z}}}.