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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