Lomp, Christian; Peña P., Alirio J. A note on prime modules. (English) Zbl 0971.16004 Divulg. Mat. 8, No. 1, 31-42 (2000). All rings are associative rings with identity and modules are left \(R\)-modules. Recall, that a module \(M\) is prime if \((0:M)=(0:N)\) for every non-zero submodule \(N\) of \(M\) and it is strongly prime if for every \(0\neq N\leq M\) and every \(m\in M\) there are elements \(n_1,\dots,n_k\in N\) such that \(\bigcap_{i=1}^k(0:n_i)\subseteq(0:m)\). Further, a module \(M\) is said to be \(B\)-prime, if \(M\) is cogenerated by each of its non-zero submodules and it is called SP-module if for every \(0\neq m\in M\) there are \(r_1,\dots,r_n\in R\) such that \(\bigcap_{i=1}^n(0:r_im)=0\). Finally a module \(M\) is called compressible if it can be embedded in any non-zero submodule of \(M\) and it is called semi-compressible if it is finitely cogenerated by any of its non-zero submodules. Let \(M\neq 0\) be such that the left annihilators of elements of \(M\) are two-sided ideals of \(R\) and let \(S=R/(0:M)\). Then (1) \(M\) is prime if and only if \(M\) is strongly prime; (2) if \(M\) is prime then \(_SM\) is an SP-module; (3) \(M\) is \(B\)-prime if and only if it is prime and \(_SM\) is cogenerated by \(S\); (4) \(M\) is semi-compressible if and only if it is prime and \(_SM\) is finitely cogenerated by \(S\); (5) \(M\) is compressible if and only if it is prime and \(_SM\) is isomorphic to a left ideal of \(S\) (Theorem 3.1). Reviewer: Ladislav Bican (Praha) Cited in 1 Document MSC: 16D80 Other classes of modules and ideals in associative algebras 16P60 Chain conditions on annihilators and summands: Goldie-type conditions 16D25 Ideals in associative algebras 16D90 Module categories in associative algebras Keywords:bounded kernel functors; annihilating sets of modules; prime modules; strongly prime modules; SP-modules; torsionfree modules; cogenerators; compressible modules PDFBibTeX XMLCite \textit{C. Lomp} and \textit{A. J. Peña P.}, Divulg. Mat. 8, No. 1, 31--42 (2000; Zbl 0971.16004) Full Text: EuDML EMIS