We know that the center of a simple ring with unity is a field. But I couldn't make an example of a ring which is not simple but its center is a field. Is it possible? Please give a hint.
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1Consider k[x,y] noncommutative polynomials. – user52045 Nov 08 '13 at 16:11
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2(^ that would be k<x,y>) – anon Nov 08 '13 at 16:17
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For example, the ring of the $2\times 2$ upper triangular matrices over $\Bbb Q$.
Having to commute with $\begin{bmatrix}1&0\\0&0\end{bmatrix}$ forces the upper right-hand coordinate of a central matrix to be zero, and commuting with $\begin{bmatrix}0&1\\0&0\end{bmatrix}$ forces the diagonal entries to be equal, so that the center is the set of matrices $\begin{bmatrix}a&0\\0&a\end{bmatrix}$ just like in the full matrix ring. So, the center is isomorphic to $\Bbb Q$.
Of course, the ideal of strictly upper triangular matrices is a nontrivial ideal.
rschwieb
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