Submodules and quotient modules of the modules with the direct summand intersection property (Q2706741)

Let \(R\) be a commutative, associative ring with identity. An \(R\)-module \(A\) has the direct summand intersection property (D.S.I.P.) if the intersection of any two direct summands of \(A\) is again a direct summand. It is known that if \(A\) has the D.S.I.P. and \(B\) is a direct summand of \(A\), then both \(B\) and \(A/B\) have the D.S.I.P. In this paper, the author examines specific types of submodules \(B\) of a module \(A\) with D.S.I.P. which are not necessarily direct summands of \(A\) and he develops necessary and sufficient conditions for which \(B\), and, in many cases, \(A/B\), have D.S.I.P. NEWLINENEWLINENEWLINEThe submodules that are investigated are \(mA\), \(A[m]\), \(m^{-1}A\) (for \(m^{-1}A\), \(A\) is considered as a submodule of another \(R\)-module \(G\)), \(F(A)\), the Frattini submodule, and \(B_A\), a \(p\)-basic submodule where \(p\) is a prime element of \(R\) and \(m\) is any element of \(R\). NEWLINENEWLINENEWLINETypical results include: NEWLINENEWLINENEWLINE(a) If \(R\) is a principal ideal domain and \(A\) is torsion, then \(A\) has D.S.I.P. if and only if \(mA\) has D.S.I.P. for every \(m\in R\). NEWLINENEWLINENEWLINE(b) If \(R\) is a P.I.D. and \(A\) is a mixed \(R\)-module with D.S.I.P., then \(A[m]\) has D.S.I.P. for each \(m\in R\). NEWLINENEWLINENEWLINE(c) For every \(m\in R\), \(A/A[m] \) has D.S.I.P. if and only if \(A[m]\) has D.S.I.P.