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?
Well, call the matrix M. Then you want that in . So just perform the calculation and see what you get,
(the x-terms act like this (commutatively) by definition).
So, what is ? What must be for ? (Put and see what happens)
So for this to be true, M^2 would have to equal M/x, and B would have to be it's own multiplicative inverse right?