For finitely generated modules over a Noetherian local ring, flatness and projectivity are equivalent.
Question 1: Let (R,m) be a Noetherian local ring, A be an Artinian R-module. So if A is flat module then A is projective module or not? If not, please give a counterexample.
We known that ” Every injective module is a direct sum of indecomposable injective module”
Question 2. Is there a similar result for flat module?
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