Semiprime and weakly compressible modules

Authors : N. DEHGHANİ, M. R. VEDADİ

Pages : 343-353

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Publication Date : 2016-04-01

Article Type : Research Paper

Abstract :An R-module M is called semiprime insert ignore into journalissuearticles values(resp. weakly compressible); if it is cogenerated by each of its essential submodules insert ignore into journalissuearticles values(resp. HomRinsert ignore into journalissuearticles values(M, N);N is nonzero for every 0 6= N ≤ MR);. We carry out a study of weakly compressible insert ignore into journalissuearticles values(semiprime); modules and show that there exist semiprime modules which are not weakly compressible. Weakly compressible modules with enough critical submodules are characterized in different ways. For certain rings R, including prime hereditary Noetherian rings, it is proved that MR is weakly compressible insert ignore into journalissuearticles values(resp. semiprime); if and only if M ∈ Coginsert ignore into journalissuearticles values(Socinsert ignore into journalissuearticles values(M); ⊕ R); and M/Socinsert ignore into journalissuearticles values(M); ∈ Coginsert ignore into journalissuearticles values(R); insert ignore into journalissuearticles values(resp. M ∈ Coginsert ignore into journalissuearticles values(Socinsert ignore into journalissuearticles values(M); ⊕ R););. These considerations settle two questions, namely Qu 1, and Qu 2, in [6, p 92].
Keywords : Krull dimension, semiprime module, singular semi Artinian ring, weakly compressible module