Ideal of Ring/Examples/Order 2 Matrices with some Zero Entries/Proof 2

Proof
Consider the mapping $\phi: R \to \R$ defined as:
 * $\forall \mathbf A \in R: \map \phi {\begin {pmatrix} x & y \\ 0 & z \end {pmatrix} } = z$

It is to be demonstrated that $\phi$ is a ring homomorphism whose kernel is $S$.

Thus:

and:

Thus by definition $\phi$ is a ring homomorphism.

By definition of $S$ itself, we have that:
 * $S \subseteq \map \ker \phi$

Then we have that:

Hence:
 * $\map \ker \phi = S$

From Kernel of Ring Homomorphism is Ideal:
 * $S is an ideal of $R$.