Exercise 24, § 1

Let E be an A-module and M, N two submodules of E. The transporter of M into N, denoted by N:M, is the set of $a \in A$ such that $aM \subset N$; it is a two-sided ideal of A, equal to $\operatorname{Ann}((M+N)/N)$.

(a) Let A be a commutative ring, M an A-module, N a sub-module of M and x an element of M; show that $N \cap Ax = (N:Ax)x$.

(b) Let A be a commutative ring and $a, b$ two ideals of A. Show that $a:b \supseteq a$ and that $(a:b)/a$ is an A-module isomorphic to $\operatorname{Hom}(A/b, A/a)$.