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let $x=\begin{bmatrix}i&0\\0&0 \end{bmatrix}$ and $y=\begin{bmatrix}0&1\\0&0 \end{bmatrix}$.

Define $A={{\begin{bmatrix}a&b\\0&c\\ \end{bmatrix}}where c\in\mathbb{R}}$

Show that A is isomorphic to $\dfrac{R<X,Y>}{((X^2+1)X),(X^2+1)Y,YX)}$

My work: Define $f:R<X,Y>\implies A$ by $f(X)=x$, $f(Y)=y$, and define $f$ to be an algebra homomorphism. Things to do: Show $(X^2+1)X$ etc. are in the kernel, which I have done.

Show $f$ is surjective, which I haven't been able to do. Finally I would need to show those elements are in fact the whole kernel in order to invoke the first isomorphism theorem

Define $A={{\begin{bmatrix}a&b\\0&c\\ \end{bmatrix}}where c\in\mathbb{R}}$

Show that A is isomorphic to $\dfrac{R<X,Y>}{((X^2+1)X),(X^2+1)Y,YX)}$

My work: Define $f:R<X,Y>\implies A$ by $f(X)=x$, $f(Y)=y$, and define $f$ to be an algebra homomorphism. Things to do: Show $(X^2+1)X$ etc. are in the kernel, which I have done.

Show $f$ is surjective, which I haven't been able to do. Finally I would need to show those elements are in fact the whole kernel in order to invoke the first isomorphism theorem

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