Question

Find all of the ideals of Q, M2(R) (the 2 x 2 matrices with entries in R) and M2(Z) (the 2 x 2 matrices with entries in Z) and determine which ideals are maximal and which ideals are prime. Please explain why the ideals are maximal and/or prime.

Answer 1

Assuming M2(R)M2(R) refers to the set of all 2×22×2 real matrices:

If AA is a rank 22 matrix, then (A)=(I)(A)=(I) is the whole ring. If AA is the zero matrix, then we get the trivial ideal. The only remaining possibility is that AA is a rank 11matrix.

By performing row operations (i.e. multiplying on the left by units) and column operations (i.e. multiplying on the right by units), we can produce any other rank 11 matrix. So, we have (1000)∈(A)(1000)∈(A) and (0001)∈(A)(0001)∈(A). Because (A)(A) is closed under addition, I∈(A)I∈(A), which means that (A)(A) is the entire ring.

Thus, the only two two-sided ideals are the trivial ideal and the ring itself.

As for nontrivial one-sided ideals, consider

{A(1000):A∈M2(R)}{(1000)A:A∈M2(R)}