Number of maximal and prime ideals

Question

Find how many prime and maximal ideals there are in the ring consisting of matrices $$M= \begin{bmatrix} a & b & c \\ 0 & a & b \\ 0 & 0 & a \\ \end{bmatrix} $$ where $a,b,c \in \mathbb{Q}$.

Answer 1

Here's a strategy that generalizes to many other similar tasks.

When asking questions about the prime and maximal ideals of a commutative ring R, we can try to simplify by modding out by nil(R), the ideal of nilpotent elements. Nothing is lost by doing this since the nilpotent elements are contained in all prime ideals, so the prime ideals of R correspond to those of R/nil(R).

It happens in this case that R/nil(R) is quite nice, and if you mod out by nil(R), the answer will be right before your eyes.

Show that if the diagonal is nonzero, such a matrix is a unit. Those obviously ant be nilpotent. Then check to see which matrices with diagonal zero are nilpotent.