isomorphism between factor ring of matrices and Z I have a commutative ring R= $\begin{pmatrix}a & b \\ 0 & a \end{pmatrix}$ (R is a 2x2 matrix, a, b $\in$ Z), I=$\begin{pmatrix}0 & b \\ 0 & 0 \end{pmatrix}$ is an ideal.
I need to prove that R/I is isomorphic to Z.
I was trying to show that I is a maximal ideal but couldn't figure it out properly. Anyone, please, help. 
or I should be prime…

We can use the first isomorphism theorem to prove this. 
Let $\phi : R \to \mathbb{Z}$ denote that mapping $\begin{pmatrix} a&b\\ 0&a \end{pmatrix} \to a$. It can be shown that $\phi$ is a homomorphism.
Then $\ker \phi = I$. And $\phi (R) = \mathbb{Z}$. 
This is all we need for the first isomorphism theorem to say $R/I \cong \mathbb{Z}$.
