Let $R$ be a ring of $2\times2$ upper triangular matrices over $\Bbb Z2$. I need to list all direct summands of left $R$-module $R$.

Question

Coz of $\Bbb Z2$, I have six upper triangular matrices in $R$. Also for left $R$-module $R$ , I have six elements just like in $R$.

But how can I get all the direct summands of it. For this, I need to find all submodules. But the problem, I find submodules with $2$ elements.

Answer 1

You don't need to find all submodules, just all summands.

Hint: The easiest way would be to compute the idempotent elements: the elements which satisfy $e^2=e$. All summands of $_RR$ will be of the form $Re$.

Of course, two idempotents may not yield distinct summands, so check what they all are.

By the way, if you think there are only six elements in the ring, count again...