Hereditary torsion theory of pseudo-regular \(S\)-systems (Q1864452)

Let \(S\) be a semigroup with 0 and 1, and let \(M\) be a right \(S\)-system. An element \(m\in M\) is called `pseudo-regular' if there exists an element \(s\not=1\) such that \(m\cdot s=m\); \(M\) is called `pseudo-regular' if all its elements are pseudo-regular. The author develops a torsion theory \(\tau_u=(U_S,\overline U_S)\) with its torsion class \(U_S\) consisting of all pseudo-regular \(S\)-systems (and \(\overline U_S\) consisting of all right \(S\)-systems with no non-zero pseudo-regular \(S\)-subsystems). For a pseudo-regular torsion theory, the structure of the corresponding quasi-filter is described. A torsion theory \((\mathcal{T,F})\) is called `stable' if \(\mathcal T\) is closed under injective hulls. The article finds conditions for a monoid \(S\) under which the torsion theory \(\tau_u=(U_S,\overline U_S)\) is stable. The special case of 0-right simple semigroups is considered.