Remark 1

Let A be a commutative ring and $\mathfrak{m}$ an ideal of A. Then $\mathfrak{m}$ is the annihilator (II, §1, No. 12, p. 219, Definition 11) of the A-module $A_s/\mathfrak{m}$. Hence, if $\mathfrak{m}$ and $\mathfrak{m}'$ are distinct ideals of A, then the A-modules $A_s/\mathfrak{m}$ and $A_s/\mathfrak{m}'$ are not isomorphic. There exists a faithful simple A-module (II, §1, No. 12, p. 219) if and only if (0) is a maximal ideal of A, that is, A is a field.