When is the subring (containing 1) of a matrix ring $M_n(k)$ over a field $k$ is local? I would be grateful for every reference concerning this matter, Thank you!
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3$\begingroup$ Do you suppose that these subrings are also $K$ algebra? If not subrings of $K$ (which may be local) may complicate things. For $n=1$, you are asking the characterization of all local subrings of a commutative field. $\endgroup$– NameMar 7, 2013 at 14:39
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$\begingroup$ Yes, I mean these subrings are also algebras over $k$. Actually I am interested in finite dimensional local rings over $k$. Any local ring $A$ can be embedded into $M_d(k)$ where $d=dim_k(A)$, but what is the minimal $n$ such that $A$ can be embedded into $M_n(k)$? Are there any estimates known? $\endgroup$– AlexanderMar 9, 2013 at 19:19
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2 Answers
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The ring of matrices $ \left( \begin{array}{cc} a & b \\\\ 0 & a \\ \end{array} \right). $
This ring is isomorphic to the algebra of dual numbers (http://en.wikipedia.org/wiki/Dual_number) which is local.
answered Mar 7, 2013 at 14:08
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The idea of @boris-novikov shows that the algebra of all upper triangular matrices such that the entries of the main diagonal are equal, is a local ring. The dimension of this algebra is $\frac{n^2-n}{2}+1$. I am wondering if one can find a local subalgebra of $M_n(k)$ whose dimension is greater than this number.
