Let $\displaystyle A$ be a commutative unital ring and $\displaystyle M$ an $\displaystyle A$-module. Suppose that $\displaystyle M\oplus A \cong A\oplus A$. Is it true that necessary $\displaystyle M\cong A$?

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- Jan 16th 2013, 07:51 AMCuruM+A = A+A implies M = A?
Let $\displaystyle A$ be a commutative unital ring and $\displaystyle M$ an $\displaystyle A$-module. Suppose that $\displaystyle M\oplus A \cong A\oplus A$. Is it true that necessary $\displaystyle M\cong A$?

- Jan 16th 2013, 11:19 AMCuruRe: M+A = A+A implies M = A?
- Jan 16th 2013, 01:08 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, 01:17 PMCuruRe: M+A = A+A implies M = A?
I really think you misunderstood the question in the OP. Maybe I had to specify that with $\displaystyle -\oplus -$ I mean the (bi)product (in the abelian category) of $\displaystyle A$-module, where $\displaystyle A$ is supposed to be commutative and unital in order to easily satisfy the IBP.