# Ideals and commutative rings Anonymous

I consider a matrix $M$ which I transform into a system of equations similar as this (but with a different $M$)

sage: M=matrix(3,3,[1,2,3,4,5,6,7,8,9])

sage: P=PolynomialRing(GF(p),M.nrows(),names="x")

sage: (vector(P.gen(i) for i in range(3))*M).list()

[x0 + 4x1 + 7x2, 2x0 + 5x1 + 8x2, 3x0 + 6x1 + 9x2]

(Example taken from another equation where $p$ is a prime).

When I try to create the ideal generated by this system, I get the following error : TypeError: R must be a commutative ring. Any idea how I can fix this?

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I do not understand what your problem is. The computation you want to perform works as far as I can tell:

sage: p = 65539
sage: M = matrix(3,3,[1,2,3,4,5,6,7,8,9])
sage: P = PolynomialRing(GF(p),M.nrows(),names="x")
sage: L =(vector(P.gen(i) for i in range(3))*M).list()
sage: P.ideal(L)
Ideal (x0 + 4*x1 + 7*x2, 2*x0 + 5*x1 + 8*x2, 3*x0 + 6*x1 + 9*x2) of Multivariate Polynomial Ring in x0, x1, x2 over Finite Field of size 65539

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My matrix is slightly different and has many zeros. Maybe this causes the ring R to be non-commutative. I will have a look. Thanks for your answer!