Let A be a domain. Assume that for any non-trivial finitely generated A -module M w

starbright49ly

starbright49ly

Answered question

2022-05-19

Let A be a domain. Assume that for any non-trivial finitely generated A-module M we have Hom A ( M , A ) { 0 }. Prove that A is a field.

Answer & Explanation

fongama33

fongama33

Beginner2022-05-20Added 12 answers

Let a A, a 0, and take f : A / ( a ) A nonzero, assuming a is not invertible.
In particular, b = f ( 1 + ( a ) ) 0 (prove it).
What can you say about a b?
osmane5e

osmane5e

Beginner2022-05-21Added 2 answers

I need help with this question too, so
a b = a f ( 1 + ( a ) ) = f ( a + a ( a ) ) = f ( 0 + ( a ) ) = 0 so a and b are non zero zero divisord. Thanks a lot, it was really easy but sometimes we can;t do easier things and can harder.

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