Proposition 13

a) If two $k$-algebras are Morita equivalent, then their centers are isomorphic $k$-algebras.

b) Two commutative $k$-algebras are Morita equivalent if and only if they are isomorphic.

c) Two $k$-algebras that are fields are Morita equivalent if and only if they are isomorphic.

d) For $i= 1,2$, let $A_i$ and $B_i$ be Morita equivalent $k$-algebras and $P_i$ an invertible $(A_i,B_i)_k$-bimodule. Set $A = A_1\otimes_kA_2,B = B_1\otimes_kB_2$, and $P = P_1\otimes_kP_2$. The $k$-algebras A and B are Morita equivalent, and P is an invertible $(A,B)_k$-bimodule.

e) If A and B are Morita equivalent $k$-algebras and $k'$ is a commutative $k$-algebra, then the $k'$-algebras $A_{(k')}$ and $B_{(k')}$ are Morita equivalent.

Assertion a) follows from Corollary 1 of VIII, p. 102, and implies b).

Let K and L be $k$-algebras that are fields, and let P be an invertible $(K,L)_k$-bimodule. The right L-vector space P is a simple module (VIII, p. 104), hence of dimension 1, so the $k$-algebras End$_L(P)$ and L are isomorphic. By VIII, p. 101, Theorem 1, the mapping $a\mapsto a_P$ from K to End$_L(P)$ is an isomorphism. Therefore, the fields K and L are isomorphic over $k$; assertion c) follows.

Under the assumptions of d), let $Q_i(i= 1,2)$ be a $(B_i,A_i)_k$-bimodule inverse to $P_i$. Denote the $(B,A)_k$-bimodule $Q_1\otimes_kQ_2$ by Q. Consider the canonical $k$-linear isomorphism $(P_1\otimes_kP_2)\otimes_k(Q_1\otimes_kQ_2)\rightarrow (P_1\otimes_kQ_1)\otimes_k$ $(P_2\otimes_kQ_2)$; when passing to the quotient, it defines a morphism

that is $(A$, A)-linear. Conversely, the inverse isomorphism $(P_1\otimes_kQ_1)\otimes_k(P_2\otimes_k$ $Q_2)\rightarrow (P_1\otimes_kP_2)\otimes_k(Q_1\otimes_kQ_2)$ defines an $(A$, A)-linear morphism $(P_1\otimes_{B_1}$ $Q_1)\otimes_k(P_2\otimes_{B_2}Q_2)\rightarrow (P_1\otimes_kP_2)\otimes_B(Q_1\otimes_kQ_2)$. These two morphisms are each other’s inverses and are thus isomorphisms. Since the $(A_i,A_i$)-bimodule $P_i\otimes_{B_i}Q_i$ is isomorphic to $A_i$, we obtain an $(A$, A)-linear isomorphism $P\otimes_B$ $Q\rightarrow A$. We likewise obtain a $(B$, B)-linear isomorphism $Q\otimes_AP\rightarrow B$, which completes the proof of d).

Under the assumptions of e), let P be an invertible $(A,B)_k$-bimodule; then $P_{(k')}$ is an invertible $(A_{(k')},B_{(k')})_{k'}$-bimodule.

Let A be a $k$-algebra, and let $e$ be an idempotent in A. The set $eAe$, endowed with the addition, multiplication, and action of $k$ induced by those of A, is a $k$-algebra with unit element $e$.