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

Example of Ideal of Ring

Let $R$ be the set of all order $2$ square matrices of the form $\begin {pmatrix} x & y \\ 0 & z \end {pmatrix}$ with $x, y, z \in \R$.

Let $S$ be the set of all order $2$ square matrices of the form $\begin {pmatrix} x & y \\ 0 & 0 \end {pmatrix}$ with $x, y \in \R$.

Then:

$R / S \cong \R$

where:

$R / S$ is the quotient ring of $R$ by $S$

$\cong$ denotes ring isomorphism.

Proof

From Ideal of Ring: Order 2 Matrices with some Zero Entries:

$S$ is an ideal of $R$

Having defined the ring homomorphism $\phi: R \to \R$:

$\forall \mathbf A \in R: \map \phi {\begin {pmatrix} x & y \\ 0 & z \end {pmatrix} } = z$

from the First Ring Isomorphism Theorem:

$\Img \phi \cong R / \map \ker \phi$

from which follows the result.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $9$: Rings: Exercise $10$