Flat modules and multiplication modules (Q1176052)

Let \(R\) be a commutative ring with identity and \(M\) a unitary \(R\)-module. Suppose \(M\) is a multiplication module, i.e. every submodule of \(M\) is of the form \(IM\) for some ideal \(I\) of \(R\). Let \(M^*=\hbox{Hom}_ R(M,R)\), and let \(T(M)=\sum_{\varphi\in M^*}\varphi(M)\) be the trace ideal of \(M\). For any non-empty subset \(S\) of an \(R\)-module \(X\), let \(\hbox{ann}(S)=\{r\in R:rS=0\}\). For each \(m\in M\), let \(D_ m=\hbox{ann(ann}\{m\})\). Also let \(D_ 0(M)=\sum_{m\in M}D_ m\). It is proved that \(T(M)=D_ 0(M)\). Moreover if (a) \(M\) contains a finite subset \(S\) such that \(\hbox{ann}(S)=\hbox{ann}(M)\); or (b) \(T(M)+\hbox{ann}(M)=R\), then \(M\) is finitely generated. Other sufficient conditions for \(M\) to be finitely generated are given in the case of \(M\) being torsionless. In addition, various sufficient conditions are given for \(M\) to be flat.