If $A \times B$ is the direct product of two rings, must $A$ necessarily be a projective $A \times B$-module?
I understand I am supposed to think of $A$ as an $A \times B$-module by identifying it with the ideal $A \times \{0\} \subset A \times B$.
2026-03-29 03:36:52.1774755412
If $A \times B$ is the direct product of two rings, must $A$ necessarily be a projective $A \times B$-module?
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I think yes, a projective module is a direct summand of a free module, and $A \oplus B \simeq A \times B$ as $A \times B$ modules. The isomorphism sends $(a, 0) + (0, b) \mapsto (a, b)$.