- #1

- 29

- 0

1. Show a ring is idempotent

2. Consider the degree one polynomial f(x) is an element of M

_{2}(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 M

_{2}(R), if any, is f(x) idempotent?

I proved part 1:

If a

^{2}= a, then a

^{2}- a = a(a-1) = 0. If a does not equal 0, then a

^{-1}exists in R and we have a-1 = (a

^{-1}a)(a-1) = a

^{-1}[a(a-1)] = a

^{-1}0 = 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?