On the prime radical and Baer's lower nilradical of modules. (Q1046856)

The author gives a generalization of the concept of \(m\)-system for modules over unitary rings. For a ring \(R\), a subset \(S\) of \(R\) is an \(m\)-system, if for each \(a,b\in S\) there exists an element \(r\in R\) such that \(arb\) is in \(S\). The complement of a prime ideal is an \(m\)-system, while, for an ideal \(I\) of \(R\), \(\sqrt I\) is the set of all elements in \(R\) for which, if they are contained in an \(m\)-system \(S\), then \(S\cap I\) is a nonempty subset of the ring. A nonempty subset \(S\) of an \(R\)-module \(M\) not containing 0 is an \(m\)-system, if for each left ideal \(A\) of \(R\) and for all submodules \(K,L\subseteq M\), if \((K+L)\cap S\) and \((K+AM)\cap S\) are nonempty, then \((K+AL)\cap S\) is nonempty. The author defines the prime radical for a submodule \(N\) of \(M\) as being the subset of those elements \(x\) in \(M\) such that if an \(m\)-system contains \(x\) then this \(m\)-system meets \(N\). The author characterizes \(m\)-systems and prime radicals and uses them for pointing out some properties as the following ones: (i) If \(M\) is an Artinian module over a PI-ring or an FBN-ring \(R\), then \(M/\text{rad}_R(M)\) is a Noetherian \(R\)-module. (ii) If \(M\) is Noetherian over a PI-ring \(R\) or an FBN-ring \(R\), such that every prime submodule is virtually maximal, then \(M/\text{rad}_R(M)\) is Artinian. (iii) The lower Baer radical of \(M\) (the set of strongly nilpotent elements of \(M\)) coincides with the prime radical for any projective \(R\)-module.