If $A\oplus B\cong A\oplus C$ and $A$ is simple, does it follow that $B\cong C$

Question

Let $R$ be a ring and $A,B,C$ be left modules over $R$, if $A\oplus B\cong A\oplus C$ and $A$ is simple (i.e. it has exactly two submodules), does it follow that $B\cong C$ ?

I believe the conclusion follows if any of $B$ or $C$ is completely reducible, my question is does it follow without any complete reduciblity assumption?

Thank you.

Answer 1

The deeper and more general statement described by rschwieb is certainly interesting, but there's also a fairly elementary proof for the particular case in the question.

Suppose $A\oplus B\to A\oplus C$ is an isomorphism, with $A$ simple. Composing with projection $A\oplus C\to A$ we get a split epimorphism $\varphi:A\oplus B\to A$ whose kernel is isomorphic to $C$.

Let $\varphi(a,b)=\varphi_A(a)+\varphi_B(b)$. Since $A$ is simple, there are two possibilities:

(1) $\varphi_A$ is an isomorphism. In this case the map $B\to\ker(\varphi)$ given by $b\mapsto (\varphi_A^{-1}\varphi_B(b),-b)$ is an isomorphism. So $C\cong B$.

(2) $\varphi_A=0$. Then $\ker(\varphi)=A\oplus\ker(\varphi_B)$, and since $\varphi_B$ is a split epimorphism, also $B\cong A\oplus\ker(\varphi_B)$. So again $B\cong C$.

Answer 2

Evans' Cancellation Theorem

Let $M$ be a left $R$ module. Then if $End(_RM)$ has stable range $1$, then $M$ is cancellable from direct sums in the category of left $R$ modules.

In your case the endomorphism ring is a division ring, and so it certainly has stable range $1$.

The original paper is here but you may find this survey of Lam's just as helpful.