Example 2

Denote by $\mathscr{B}_A,\mathscr{B}_B$, and $\mathscr{B}_P$ the sets of two-sided ideals of A, two-sided ideals of B, and $(A,B)_k$-sub-bimodules of P, respectively.

a) The mapping $\mathfrak{b}\mapsto P\mathfrak{b}$ is an isomorphism of ordered sets from $\mathscr{B}_B$ to $\mathscr{B}_P$; the inverse isomorphism sends an $(A,B)_k$-sub-bimodule $P'$ of P to the two-sided ideal of B consisting of the elements $b$ such that $Pb\subset P'$.

b) The mapping $\mathfrak{a}\mapsto \mathfrak{a}P$ is an isomorphism of ordered sets from $\mathscr{B}_A$ to $\mathscr{B}_P$; the inverse isomorphism sends an $(A,B)_k$-sub-bimodule $P'$ of P to the two-sided ideal of A consisting of the elements $a$ such that $aP\subset P'$.

Indeed, let J be a left ideal of B and $P'=$ PJ. Then $P'$ is an A-submodule of P and, by Example 1, the ideal J consists of the elements $b$ of B such that $Pb\subset P'$. Moreover, $P'$ is an $(A,B)_k$-sub-bimodule of P if and only if J is a two-sided ideal of B. Thus a) follows from loc. cit.

Assertion b) follows from assertion a) applied to the invertible $(B^o,A^o)_k$-bimodule P.