Goldie absolute direct summand rings and modules. (English) Zbl 1438.16014

Summary: In the present paper, we introduce and study Goldie ADS modules and rings, which subsume two generalizations of Goldie extending modules due to E. Akalan et al. [Commun. Algebra 37, No. 2, 663–683 (2009; Zbl 1214.16005); Corrigendum 38, No. 12, 4747–4748 (2010); Corrigendum 41, No. 5, 2005 (2013)] and ADS-modules due to A. Alahmadi et al. [J. Algebra 352, No. 1, 215–222 (2012; Zbl 1256.16005)]. A module \(M\) will be called a Goldie ADS module if for every decomposition \(M = S\oplus T\) of \(M\) and every complement \(T'\) of \(S\), there exists a submodule \(D\) of \(M\) such that \(T'\beta D\) and \(M = S\oplus D\). Various properties concerning direct sums of Goldie ADS modules are established.


16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16D40 Free, projective, and flat modules and ideals in associative algebras
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16N20 Jacobson radical, quasimultiplication
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