Structure of some Noetherian SI rings
Motivated by the study of right SI rings (those rings for which every right singular module is injective), \textit{P. F. Smith} introduced right CDPI rings [Can. J. Math. 31, 93-111 (1979; Zbl 0412.16012)]. Namely, a ring is right CDPI if every cyclic right module is a direct sum of a projective and an injective module. In that same paper, an example of a right SI ring that is not right CDPI was given. Many years later, it was shown that all right CDPI rings are right SI [\textit{B. L. Osofsky, P. F. Smith}, J. Algebra 139, No. 2, 342-354 (1991; Zbl 0737.16001)]. One of the purposes of the paper under review is to give conditions under which a right SI ring is right CDPI. This result is achieved by providing a full characterization of right CDPI rings as ring-direct sums of subrings satisfying various technical conditions (Theorem 6). This theorem may be viewed as a fine tuning of the characterization of right SI rings given by \textit{K. R. Goodearl} [in Mem. Am. Math. Soc. 124 (1972; Zbl 0242.16018)] for the specific family of right CDPI rings. The paper concludes with a nice section devoted to examples to illustrate the feasibility of the different types of subrings that appear in the structure described in Theorem 6.
scientific article; zbMATH DE number 1867984
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