Matrix algebra over a polynomial ring

We consider the algebra , where P is the polynomial ring in indeterminate x over the field . We take the matrix having in its (i,j)-th position.

> K := GaloisField(5);
> P<x> := PolynomialAlgebra(K);
> M := MatrixAlgebra(P, 5);
> a := M ! [x^i + x^j: i, j in [1..5]];
> a;
[      2*x   x^2 + x   x^3 + x   x^4 + x   x^5 + x]
[  x^2 + x     2*x^2 x^3 + x^2 x^4 + x^2 x^5 + x^2]
[  x^3 + x x^3 + x^2     2*x^3 x^4 + x^3 x^5 + x^3]
[  x^4 + x x^4 + x^2 x^4 + x^3     2*x^4 x^5 + x^4]
[  x^5 + x x^5 + x^2 x^5 + x^3 x^5 + x^4     2*x^5]
> ElementaryDivisors(a);
[
    x,
    x^3 + 3*x^2 + x
]

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