## 22.14 Projective modules over algebras

In this section we discuss projective modules over algebras analogous to Algebra, Section 10.77. This section should probably be moved somewhere else.

Let $R$ be a ring and let $A$ be an $R$-algebra, see Section 22.2 for our conventions. It is clear that $A$ is a projective right $A$-module since $\mathop{\mathrm{Hom}}\nolimits _ A(A, M) = M$ for any right $A$-module $M$ (and thus $\mathop{\mathrm{Hom}}\nolimits _ A(A, -)$ is exact). Conversely, let $P$ be a projective right $A$-module. Then we can choose a surjection $\bigoplus _{i \in I} A \to P$ by choosing a set $\{ p_ i\} _{i \in I}$ of generators of $P$ over $A$. Since $P$ is projective there is a left inverse to the surjection, and we find that $P$ is isomorphic to a direct summand of a free module, exactly as in the commutative case (Algebra, Lemma 10.77.2).

We conclude

the category of $A$-modules has enough projectives,

$A$ is a projective $A$-module,

every $A$-module is a quotient of a direct sum of copies of $A$,

every projective $A$-module is a direct summand of a direct sum of copies of $A$.

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