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Are there some programs with packages that help to list all zero-divisors of the ring we desired? especially for the ring of Gaussian integers modulo $n$, $\mathbb{z}_n[i]$.

Also want to find which of the product of ring elements will result in 0. I wish to get the details if any. I am new to programming languages, this is because I want to get help to verify my theoretical results.

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I think that what you are looking for is a Computer Algebra System (CAS). You could find a list in Wikipedia.

I don't know which one suits your particular interest, although Singular seems to be a good candidate.

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