Centralizer in a matrix algebra over commutative polynomials - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T20:21:44Z http://mathoverflow.net/feeds/question/116821 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116821/centralizer-in-a-matrix-algebra-over-commutative-polynomials Centralizer in a matrix algebra over commutative polynomials spelas 2012-12-19T21:46:04Z 2012-12-19T21:46:04Z <p>Let $A=M_n(F[x_{ij}\mid 1\leq i,j\leq n])$ be the matrix algebra over commutative polynomials in $n^2$ variables $x_{11},\dots,x_{nn}$ over a (nice enough) field $F$. I would like to know what is the centralizer of the matrix $x=(x_{ij})$ in $A$.</p> <p>If we consider $x$ as an element of $M_n(F(x_{ij}\mid 1\leq i,j\leq n))$, it is not difficult to show that its centralizer consists of polynomials in $x$ with coefficients in $F(x_{ij}\mid 1\leq i,j\leq n)$. Does the centralizer of $x$ in $A$ consists of polynomials in $x$ with coefficients in $F[x_{ij}\mid 1\leq i,j\leq n]$?</p>