Inverting an element (in Computer Algebra System)

Question

I need to compute Groebner Basis over rings, which are obtained by inverting an element. In other words, I would like to localize the ring at this (non-zero) element. The base coefficient ring need not be a field.

For example: $\mathbb{Z}\left[\frac{1}{2}\right][x,y]$.

I know how to invert $x$: $$\mathbb{Z}\left[\frac{1}{x}\right][x,y]\cong \frac{\mathbb{Z}[x,y][t]}{\langle tx-1\rangle}.$$ But this trick doesnt work for arbitrary elements!

I am currently working with Singular and Macaulay2. I would prefer a solution within the ambit of these CASs. Any pointers?

-- Mike

Answer 1

Note that $$\mathbb{Z}\left[\frac{1}{2}\right][x,y]\cong \frac{\mathbb{Z}[t]}{\langle 2t-1\rangle}[x,y].$$

In Singular you can try:

ring r = integer, (x,y,t),dp; qring q = std(2t-1); q;

I am sure something similar can be done in Macaulay2. However MAGMA provides a complete comprehensive solution. You can try that as well.