# Computer algebra system that test zero divisors in a quotient algebra

I have an algebra $A$ over a Noetherian ring and an ideal $I=(x,y)$, where $x,y \in A$. I need to examine whether a polynomial $h \in A$ is a zero divisor in $A/I$ or not.

Is there a computer algebra system can do that?

Thanks for help.

• You'll need to give more information about $A$. What kind of presentation of $A$ do you have? – Neil Strickland Apr 20 '16 at 12:40
• @NeilStrickland $A$ is a subalgebra of $R[a_1,a_2]$, generated by $a_1^2, a_2^2, a_1 a_2$. – user279941 Apr 20 '16 at 12:48
• What is $R$? Is $R[a_1,a_2]$ a polynomial ring? Or a free associative algebra? – Max Horn Apr 20 '16 at 16:55
• @MaxHorn say $R=Z$. Yes, it is a polynomial ring in two variables. – user279941 Apr 20 '16 at 20:19

You can also do this in Macaulay2 http://www.math.uiuc.edu/Macaulay2/

Here's the commands:

A = ZZ[a,b,c]/ideal(a*b-c^2);
I = ideal(x,y);
(ker(matrix{{h}}**(A/I))) == 0


To briefly explain: you're constructing the multiplication by $h$ map over $A/I$ and computing its kernel and testing if it's the 0 module. So you'll get true if $h$ is a nonzerodivisor.

MAGMA can do it for sure in case of $A=\mathbb{Z}[a_1^2,a_1a_2,a_2^2]$. But might have problems if the coefficients are more complicated rings.

However you should see $A$ as $\mathbb{Z}[b_1,b_2,b_3]/(b_1b_3-b_2^2)$ (where $b_1=a_1^2$ etc.) and hence, define $A/(x,y)$ directly as $B:=\mathbb{Z}[b_1,b_2,b_3]/(b_1b_3-b_2^2,x,y)$

I also guess, it will always be a quick computation, so that you can use MAGMA's online-calculator (in case you don't have other access to MAGMA).