On direct sums of modules which satisfy generalizations of injectivity (Q2710268)

This paper is a survey of some results regarding generalizations of injectivity. Most of the results quoted concern conditions for direct sums of generalized injective modules (mostly of copies of a single fixed module \(M\)) to be generalized injective again. Given a module property \(P\), a module \(M\) is said to be \(\Sigma\)-\(P\) if the direct sum \(M^{(I)}\) (\(I\) some index set) satisfies \(P\) again.NEWLINENEWLINENEWLINEThe following are some of the results quoted:NEWLINENEWLINENEWLINE\(\bullet\) An injective \(R\)-module \(E\) is \(\Sigma\)-injective if and only if \(R\) satisfies ACC on the \(t\)-annihilator left ideals [\textit{C. Faith}, Nagoya Math. J. 27, 179-191 (1966; Zbl 0154.03001)].NEWLINENEWLINENEWLINE\(\bullet\) If \(Q\) is a quasi-injective module, then \(Q\) is \(\Sigma\)-quasi-injective if and only if \(E(Q)\) is \(\Sigma\)-injective [\textit{A. Cailleau} and \textit{G. Renault}, C. R. Acad. Sci., Paris, Sér. A 270, 1391-1394 (1970; Zbl 0197.31201)].NEWLINENEWLINENEWLINE\(\bullet\) An \(R\)-module \(M\) is \(\Sigma\)-quasi-injective if and only if \(M\) is \(\Sigma\)-continuous [\textit{L. Jeremy}, Can. Math. Bull. 17, 217-228 (1974; Zbl 0301.16024)].NEWLINENEWLINENEWLINE\(\bullet\) A ring \(R\) is a left \(\Sigma\)-extending ring if and only if every module is the direct sum of a projective module and a singular module [\textit{K. Oshiro}, Hokkaido Math. J. 13, 310-338 (1984; Zbl 0559.16013)].NEWLINENEWLINENEWLINE\(\bullet\) A pure-injective module \(M\) is \(\Sigma\)-pure-injective if and only if any direct product of copies of \(M\) is a direct sum of indecomposable modules [\textit{B. Zimmermann-Huisgen}, Proc. Am. Math. Soc. 77, 191-197 (1979; Zbl 0441.16016)].NEWLINENEWLINEFor the entire collection see [Zbl 0940.00021].