Definition 2

Two $k$-algebras A and B are called Morita equivalent if there exists an invertible $(A,B)_k$-bimodule. Rings A and B are called Morita equivalent if the $\mathbf{Z}$-algebras A and B are Morita equivalent.

Two isomorphic $k$-algebras are Morita equivalent. If two $k$-algebras are Morita equivalent, then their opposite algebras are Morita equivalent.

Let P be an invertible $(A, B)_k$-bimodule and Q an inverse of P. Then Q is an invertible $(B, A)_k$-bimodule, and it has P as inverse. Moreover, viewed as a $(B^\circ, A^\circ)_k$-bimodule, P is invertible and has the $(A^\circ, B^\circ)_k$-bimodule Q as inverse.