Let $(S,+,\cdot)$ be the commutative and unitary ring given by
$$S=\left\{\left(\begin{array}{cc}a&b\\0&a \end{array}\right)\mid a,b\in\mathbb{R}\right\}$$
Find all the ideals.
Are they all prime ideals?
I found that one ideal is
$$I=\left(\begin{array}{cc}0&b\\0&0 \end{array}\right)$$ and it's not a prime ideal.
How can you know that there are or are not more ideals?
Best Answer
Note that the elements of the ideal $I$ you found are exactly the non invertible elements of the ring. It follows that if $J$ is a proper ideal then $J\subseteq I$. Also, if $J$ contains a nonzero element $\left(\begin{array}{cc}0&b\\0&0 \end{array}\right)$, then for each $c\in\mathbb{R}$ we have:
$\left(\begin{array}{cc}0&c\\0&0 \end{array}\right)=\left(\begin{array}{cc}0&b\\0&0 \end{array}\right)\left(\begin{array}{cc}\frac{c}{b}&1\\0&\frac{c}{b} \end{array}\right)\in J$
And so $J=I$ in that case. So it follows that $I$ is the only nontrivial ideal.
And by the way, $I$ is a maximal ideal, so in particular it is prime.