Some results on SI-rings.

*(English)*Zbl 0830.16005A ring \(R\) is called a right \(SI\)-ring if every singular right \(R\)-module is injective. Right \(SI\)-rings were introduced and studied by K. R. Goodearl [Mem. Am. Math. Soc. 124 (1972; Zbl 0242.16018)] and he proved that a ring \(R\) is a right \(SI\)-ring if and only if \(R\) is right nonsingular and \(R=K\oplus R_1\oplus\dots\oplus R_n\) for some positive integer \(n\), ring \(K\) with \(K/\text{Soc} (K_K)\) semisimple and rings \(R_i\) \((1\leq i\leq n)\) each Morita equivalent to a right \(SI\)-domain. Subsequently B. L. Osofsky and the reviewer [J. Algebra 139, No. 2, 342-354 (1991; Zbl 0737.16001)] showed that a ring \(R\) is right \(SI\) if and only if every cyclic singular right \(R\)-module is injective.

In this note the authors prove that a ring \(R\) is right \(SI\) if and only if every cyclic semiprimitive singular right \(R\)-module is injective. (A module \(M\) is called semiprimitive if \(\text{Rad } M=0\).) A ring \(R\) is called a right \(CS\)-ring if every right ideal is an essential submodule of a direct summand of \(R_R\). The main theorem states: a ring \(R\) is a ring direct sum of a semiprimary \(SI\)-ring and a right \(CS\) right \(SI\)- ring with zero right socle if and only if every cyclic semiprimitive right \(R\)-module is a direct sum of a projective module and an injective module. In addition a ring \(R\) is the direct sum of a semiprimary \(SI\)- ring and an \(SI\)-ring with zero right (and left) socle if and only if every finitely generated semiprimitive right (or left) \(R\)-module is a direct sum of a projective module and an injective module. Finally a ring \(R\) is the ring direct sum of a semisimple ring and a right \(SI\)-domain if and only if every cyclic semiprimitive right \(R\)-module is projective or injective.

In this note the authors prove that a ring \(R\) is right \(SI\) if and only if every cyclic semiprimitive singular right \(R\)-module is injective. (A module \(M\) is called semiprimitive if \(\text{Rad } M=0\).) A ring \(R\) is called a right \(CS\)-ring if every right ideal is an essential submodule of a direct summand of \(R_R\). The main theorem states: a ring \(R\) is a ring direct sum of a semiprimary \(SI\)-ring and a right \(CS\) right \(SI\)- ring with zero right socle if and only if every cyclic semiprimitive right \(R\)-module is a direct sum of a projective module and an injective module. In addition a ring \(R\) is the direct sum of a semiprimary \(SI\)- ring and an \(SI\)-ring with zero right (and left) socle if and only if every finitely generated semiprimitive right (or left) \(R\)-module is a direct sum of a projective module and an injective module. Finally a ring \(R\) is the ring direct sum of a semisimple ring and a right \(SI\)-domain if and only if every cyclic semiprimitive right \(R\)-module is projective or injective.

Reviewer: P.F.Smith (Glasgow)

##### MSC:

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |

16D50 | Injective modules, self-injective associative rings |

16D40 | Free, projective, and flat modules and ideals in associative algebras |

16D90 | Module categories in associative algebras |