Definition 1

Let A and B be $k$-algebras and P an $(A,B)_k$-bimodule. We say that P is invertible if there exists a $(B,A)_k$-bimodule Q such that $P\otimes_BQ$ is isomorphic to $_sA_d$ and $Q\otimes_AP$ to $_sB_d$. Such a bimodule Q is called an inverse of P.

Let A and B be $k$-algebras, and let P be an invertible $(A,B)_k$-bimodule. Let C be a $k$-algebra and $P'$ an invertible $(B,C)_k$-bimodule. Finally, let Q and $Q'$ be inverse bimodules of P and $P'$, respectively. By the associativity of the tensor product (II, §3, No. 8, p. 258, Proposition 8) and Proposition 4 of II, §3, No. 4, p. 249, the $(C,A)_k$-bimodule $Q'\otimes_BQ$ is an inverse of the $(A,C)_k$-bimodule $P\otimes_BP'$, so that $P\otimes_BP'$ is an invertible $(A,C)_k$-bimodule.

Hence, the relation

“A and B are $k$-algebras, and there exists an invertible $(A,B)_k$-bimodule” is an equivalence relation.