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

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- Jan 16th 2013, 08:51 AMCuruM+A = A+A implies M = A?
Let be a commutative unital ring and an -module. Suppose that . Is it true that necessary ?

- Jan 16th 2013, 12:19 PMCuruRe: M+A = A+A implies M = A?
- Jan 16th 2013, 02:08 PMHallsofIvyRe: M+A = A+A implies M = A?
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.

- Jan 16th 2013, 02:17 PMCuruRe: M+A = A+A implies M = A?