Alahmadi, Adel; Jain, S. K.; Leroy, André ADS modules. (English) Zbl 1256.16005 J. Algebra 352, No. 1, 215-222 (2012). The notion of modules with the ‘absolute direct summand property’ (ADS, for short) was introduced by L. Fuchs [in Infinite Abelian groups. Vol. I. Pure and Applied Mathematics 36. New York-London: Academic Press (1970; Zbl 0209.05503)]. Fuchs called a right module \(M\) right ADS if for every decomposition \(M=S\oplus T\) of \(M\) and every complement \(C\) of \(S\) we have \(M=S\oplus C\). For example, if \(R\) is an Abelian ring then \(R\) as a right \(R\)-module is ADS. Every right quasi-continuous module is ADS, but the converse need not be true. If a right ADS module is also right CS, then it is right quasi-continuous. The paper under review shows that if \(R\) is a simple ring such that \(R_R\) is ADS, then either \(R\) is right self-injective or indecomposable as a right \(R\)-module. W. D. Burgess and R. Raphael [in Ring theory. Proceedings of the biennial Ohio State-Denison mathematics conference, 1992. Singapore: World Scientific. 137-148 (1993; Zbl 0853.16008)] claimed that an example can be constructed of a finite dimensional module over a finite dimensional algebra which has no ADS hull. This paper shows that under certain conditions such an ADS hull does exist. A right module \(M\) is called completely ADS if each of its subfactors is ADS. Among other results it is shown that if \(M\) is a semiperfect module with a completely ADS projective cover, then \(M=S\oplus T\) where \(S\) is semisimple and \(T\) is a sum of local modules. Reviewer: Ashish K. Srivastava (Saint Louis) Cited in 2 ReviewsCited in 20 Documents 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 16D80 Other classes of modules and ideals in associative algebras Keywords:ADS modules; injective modules; quasi-continuous modules; direct sums; absolute direct summand property; ADS hulls Citations:Zbl 0209.05503; Zbl 0853.16008 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Burgess, W. D.; Raphael, R., On modules with the absolute direct summand property, (Ring Theory. Ring Theory, Granville, OH, 1992 (1993), World Sci. Publ.: World Sci. Publ. River Edge, NJ), 137-148 · Zbl 0853.16008 [2] Fuchs, L., Infinite Abelian Groups, vol. I, Pure Appl. Math., Ser. Monogr. Textb., vol. 36 (1970), Academic Press: Academic Press New York, San Francisco, London · Zbl 0209.05503 [3] Goodearl, K. R., Von Neumann Regular Rings (1991), Krieger Publishing Company: Krieger Publishing Company Malabar, FL · Zbl 0749.16001 [4] Goel, V. K.; Jain, S. K., \(π\)-injective modules and rings whose cyclics are \(π\)-injective, Comm. Algebra, 6, 59-73 (1978) · Zbl 0368.16010 [5] Kasch, F., Modules and Rings (1982), Academic Press · Zbl 0523.16001 [6] Mohammed, S. H.; Müller, B. J., Continuous and Discrete Modules, London Math. Soc. Lecture Note Ser., vol. 147 (1990), Cambridge University Press: Cambridge University Press New York · Zbl 0701.16001 [7] Wisbauer, R., Foundations of Module and Ring Theory (1991), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers Reading · Zbl 0746.16001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.