Direct product of projective modules

Question

For any nonempty family like $\{ {M_\alpha}\}_{\alpha \in I}$ of $R$-modules we know that if $\prod_{\alpha \in I} M_\alpha$ is a projective $R$-module,then for all $\alpha \in I$, $\{ {M_\alpha}\}_{\alpha \in I}$ is projective too. Is the converse true? If not give me a counter-example.

Any comments are welcome.

Answer 1

A theorem by S. U. Chase states that for a ring $R$ the following conditions are equivalent:

1. $R$ is left perfect and right coherent

2. every product of projective left $R$-modules is projective

It's Theorem 3.3 in S. U. Chase, Direct products of modules, Transactions of the American Mathematical Society, 97 (1960), 457–473.

(“Finitely related” is nowadays more commonly referred to as “finitely presented”.)

So you just need to take a non left perfect ring and you have a counterexample: a suitable direct power of the regular module won't be projective.