My professor gave us this query at the end of class, it contained two parts.

1. Show a ring is idempotent

2. Consider the degree one polynomial f(x) is an element of M2(R)[x] given by

f(x) = [0 1

______0 0]x + B

(so f(x) = the matrix []x + B).

For which B is an element of M2(R), if any, is f(x) idempotent?

I proved part 1:

If a2 = a, then a2 - a = a(a-1) = 0. If a does not equal 0, then a-1 exists in R and we have a-1 = (a-1a)(a-1) = a-1[a(a-1)] = a-10 = 0, so a-1 = 0 and a = 1. Thus 0 and 1 are hte only two idempotent elements in a division ring.

Part 2 I simply have no idea on though. How do I do this with a matrix?