# Proving that an element of a ring is regular

Suppose that I have explicit polynomials $f_1, f_2, \ldots, f_m$ in the variables $x_1, \ldots, x_n$ with coefficients in $\Bbb Z$ and no constant term. Then, for any prime $p$, I would like to compute whether or not $x_1$ is a zero-divisor in the ring

$$R = \Bbb F_{p}[[x_1,\ldots,x_n]]/(f_1,\ldots,f_m).$$

Is this something I can do with one of the available commutative algebra packages? Ideally, I would like a simple way to certify that $x_1$ will not be a zero-divisor in $R$ for any prime $p$ outside a finite list (if that is true, of course).

-
I never used it but I think Singular (package) is your best bet. It is free btw... –  Sergio Parreiras Dec 9 '13 at 3:11
Macaulay2 can handle $F_p[x_1,\dots, x_n]/ (f_i's)$, but I am not sure if there is a computer algebra package can handle completions. I heard that Singular as mentioned above can handle local rings, but I am not sure about completion. –  Youngsu Dec 9 '13 at 4:10
@Youngsu Why need completions as $A\to\hat{A}$ is faithfully flat (in this case)? –  user26857 Dec 9 '13 at 8:25
@YACP: Yes. You are right. Thanks for pointing it out. –  Youngsu Dec 9 '13 at 8:45