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ADS modules. (English) Zbl 1256.16005

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.

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
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[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
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