In what follows all rings are commutative with identity. A module \(M\) over a ring \(R\) is called a multiplication module if each submodule \(N\) of \(M\) is of the form \(IM\) for some ideal \(I\) of \(R\). There is a close relationship between multiplication modules and projectivity if the modules are finitely generated. For example, it is known that each finitely generated projective ideal is a multiplication ideal and, in the opposite direction, if the annihilator of a finitely generated multiplication module \(M\) is generated by an idempotent then \(M\) is projective [see the author, H. J. Mustafa and M. S. Abdullah, Period. Math. Hung. 20, No. 1, 57-63 (1989; Zbl 0631.13008) and P. F. Smith, Arch. Math. 50, No. 3, 223-235 (1988; Zbl 0615.13003)].
Here the author looks at the relationship without the finite generation assumption. His main result is that for a projective \(R\)-module \(M\) the following conditions are equivalent: (1) \(M\) is a multiplication module, (2) the endomorphism ring of \(M\) is commutative, (3) \(M\) is locally cyclic, (4) \(T(M)\subseteq\theta(M)\), where \(T(M)\) is the trace of \(M\) while \(\theta(M)=\sum_{a\in M}\{r\in R:rM\subseteq Ra\}\). The author also gives a useful alternative description of the trace of a projective module using the double annihilators of its elements.