Modules with comparability (Q2770607)

Rings with comparability were introduced by \textit{M. Ferrero} and \textit{A. Sant'Ana} [Can. Math. Bull. 42, No. 2, 174-183 (1999; Zbl 0949.16030)] as a class of rings which properly contains right distributive rings. In this paper modules with comparability are investigated. Let \(R\) be a ring and \(P\) be a completely prime ideal which is contained in the Jacobson radical \(J(R)\). Then \(P\) is called (right) admissible if \(S=R\setminus P\) is a right Ore set. It was shown by \textit{M. Ferrero} and \textit{A. Sant'Ana} in the above quoted paper, that if \(R\) is a ring with \(P\)-comparability, then \(P\) is an admissible ideal. Let \(M\) be a right \(R\)-module and \(P\) an admissible ideal of \(R\). Then \(M\) is said to be an \(R\)-module with \(P\)-comparability if for every \(x,y\in M\) either \(xR\subseteq yR\) or \((xR)S^{-1}=(yR)S^{-1}\), where \(S=R\setminus P\). For a module \(M\), a submodule \(L\) of \(M\) is called a waist if \(L\subseteq N\) or \(N\subseteq L\) for every submodule \(N\).NEWLINENEWLINENEWLINEAssume that \(M\) is a right \(R\)-module with \(Q\)-comparability, where \(Q\) is a completely prime ideal of \(R\) contained in \(J(R)\) which is a waist of \(R\) as right ideal and \(MQ\neq 0\). Then it is shown that for any \(x\in M\setminus MQ\), there is a one-to-one correspondence between submodules of \(MQ\) and right ideals of \(R\) which are contained in \(Q\) and contain the annihilator of \(x\). Also it is proved that a submodule \(L\) of \(MQ\) is a completely prime (resp. prime, semiprime) submodule of \(M\) if and only if the corresponding right ideal of \(R\) is completely prime (resp. prime, semiprime) for any \(x\notin MQ\).