Inverting an element (in Computer Algebra System) 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

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.