M+A = A+A implies M = A?

Let be a commutative unital ring and an -module. Suppose that . Is it true that necessary ?

The answer is affirmative and contained in these beautiful notes (click) by Brian Conrad. A proof an be find in Weibel's K-Book (click).

It doesn't really require a very complicated proof. In any ring, not just "commutative" or "unary" (in fact, since there is no mention of multiplication, you don't even need a ring), we have additive inverses. Adding the additive inverse of A to both sides immediately results in M= A.

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Originally Posted by HallsofIvy

It doesn't really require a very complicated proof. In any ring, not just "commutative" or "unary" (in fact, since there is no mention of multiplication, you don't even need a ring), we have additive inverses. Adding the additive inverse of A to both sides immediately results in M= A.

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I really think you misunderstood the question in the OP. Maybe I had to specify that with I mean the (bi)product (in the abelian category) of -module, where is supposed to be commutative and unital in order to easily satisfy the IBP.