# Finding the kernel of a matrix in a non-integral domain

I have been trying to find the kernel of the matrix in a quotient, for example. If we have the following quotient ring in sage:

```
R.<t> = PolynomialRing(GF(3),'t')
I = R.ideal([t^3])
S = R.quotient_ring(I);
```

and if I try to find the kernel of the matrix:

```
E = Matrix(S, ([[0+a*t+b*t^2, 1+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2],
[0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2],
[0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2, 1+a*t+b*t^2],
[0+a*t+b*t^2, 0+a*t+b*t^2, 0+a*t+b*t^2,0+a*t+b*t^2]]))
E.kernel()
```

It gives me the following error: NotImplementedError.

I guess this is because F3[x]/(x^3) is not an integral domain but I would like a way around it.

Thanks in advance.

How do you define

`a`

and`b`

?a and b are in F3. yes, i should have mentioned.

Did you fix them or are them indeterminates ("variables") ?