I just encountered one statement out of blue when reading section 20 of 'local representation theory ' by Alperin.
Every module is a homomorphic image of a projective module.
Not sure how to see this. Any hint would be appreciated! Thank you!
2026-03-28 01:12:51.1774660371
Homomorphic image of a projective module
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We know that every $R-$module $M$ is a quotient of a free $R-$module using the typical surjective map $$ \bigoplus_{x\in M} Rx\to M\to 0.$$ A free module is projective, so we are done.