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.
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$\begingroup$ You'll need to give more information about $A$. What kind of presentation of $A$ do you have? $\endgroup$– Neil StricklandApr 20, 2016 at 12:40
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$\begingroup$ @NeilStrickland $A$ is a subalgebra of $R[a_1,a_2]$, generated by $a_1^2, a_2^2, a_1 a_2$. $\endgroup$– user279941Apr 20, 2016 at 12:48
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$\begingroup$ What is $R$? Is $R[a_1,a_2]$ a polynomial ring? Or a free associative algebra? $\endgroup$– Max HornApr 20, 2016 at 16:55
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$\begingroup$ @MaxHorn say $R=Z$. Yes, it is a polynomial ring in two variables. $\endgroup$– user279941Apr 20, 2016 at 20:19
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2 Answers
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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.
answered Jun 19, 2016 at 16:24
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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).
answered May 20, 2016 at 14:07