Exercise 4, § 6

Let A and B be rings and P an invertible $(A$, B)-bimodule. Denote the set of two-sided ideals of A (resp. B) by $\mathscr{B}_A$ (resp. $\mathscr{B}_B$) and the isomorphism of ordered sets defined in Proposition 6 (VIII, p. 105) by $f:\mathscr{B}_A\rightarrow \mathscr{B}_B$.

a) Let $\mathfrak{a}$ be an ideal of A. Show that the rings $A/\mathfrak{a}$ and $B/f(\mathfrak{a})$ are Morita equivalent. b) Let $\mathfrak{a}$ and $\mathfrak{a}'$ be ideals of A. Show that $f(\mathfrak{a}\mathfrak{a}') =f(\mathfrak{a})f(\mathfrak{a}')$.

c) Show that an ideal $\mathfrak{a}$ of A is nilpotent if and only if the ideal $f(\mathfrak{a})$ of B is nilpotent. d) Let M be a B-module. Prove that the trace ideal of M (VIII, p. 80) is the image under $f$ of the trace ideal of $P\otimes_BM$.