Example 3

Let A be a principal ideal domain and L a nonzero, finitely generated, free A-module. Let B be the endomorphism ring of L; then L is an invertible $(A,B)_{\mathbf{Z}}$-bimodule, and the rings A and B are Morita equivalent. By Morita’s theorem, Proposition 10, a) (VIII, p. 109), and the structure theorem for finitely generated A-modules (VII, §4, No. 4, p. 19, Theorem 2), every finitely generated B-module is isomorphic to $\oplus^m_{i=1}(L/\mathfrak{a}_iL)$, where $m$ is a natural number and the $\mathfrak{a}_i$ are ideals of A satisfying $\mathfrak{a}_1\subset \mathfrak{a}_2\subset \cdots \subset \mathfrak{a}_m$ and $\mathfrak{a}_m\not= A$; the integer $m$ and the ideals $\mathfrak{a}_i$ are uniquely determined. By Proposition 6 of VIII, p. 105, every two-sided ideal of B is of the form $dB$, where $d$ is an element of A.