Progenerator

Remarks

A module $P$ is a generator if and only if $R$ is a direct summand of $P^n$ for some $n \ge 1$. Example: $P = R^n$ with $n \ge 1$ is free of finite rank, hence finitely generated and projective, and $R$ is a direct summand of it, so $P$ is a progenerator. Non-example: let $k$ be a field, $R = k \times k$ and $P = k \times 0 = R(1, 0)$. It is finitely generated and projective, as a direct summand of $R$. It is not a generator: every map $P \to 0 \times k$ is zero, since it sends $(1, 0) = (1, 0)^2$ to $(1, 0) \cdot m = 0$; so $0 \times k$ is not a quotient of a sum of copies of $P$. Over a commutative ring $R$, a finitely generated projective module is a generator if and only if it is faithful, that is, its rank is positive at every prime ideal. The ring $\End_R(P)$ of a progenerator $P$ is Morita equivalent to $R$.