Exercise 5, § 3

Let A be a ring.

a) Let M, N be monogenous A-modules. Prove that the following properties are equivalent:

(i) The A-modules M and N are isomorphic.

(ii) For every generator $m$ of M, there exists a generator $n$ of N that has the same annihilator as $m$.

(iii) There exist a generator of M and a generator of N that have the same annihilator.

b) Let $\mathfrak{a},\mathfrak{b}$ be (left) ideals of A. The A-modules $A/\mathfrak{a}$ and $A/\mathfrak{b}$ are isomorphic if and only if there exists an element $a$ of A such that $Aa+\mathfrak{b}= A$ and $\mathfrak{a}$ is the set of elements $x$ of A such that $xa\in \mathfrak{b}$ (use a)).

c) Let $\mathfrak{m}$ and $\mathfrak{n}$ be maximal left ideals of A. The simple A-modules $A/\mathfrak{m}$ and $A/\mathfrak{n}$ are isomorphic if and only if there exists an element $a$ of A$-\mathfrak{n}$ such that $\mathfrak{m}a\subset \mathfrak{n}$. d) Deduce from b) that the set of quotient A-modules of the A-module $A_s$ isomorphic to a given module has cardinal less than Card(A).