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Some results on SI-rings. (English) Zbl 0830.16005
A 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.

##### 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
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