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The question above.

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I know that the spell checker in your browser will not be able to see the typo in your title, but presumably you can? – Mariano Suárez-Alvarez Sep 28 '10 at 12:31
I fixed the typo. – Timothy Chow Sep 28 '10 at 14:19

Sort of yes, because there is a better algorithm for rings, known under umbrella name of Grobner Basis. It works the nicest for commutative algebras, but there are various versions for algebras over Euclidian domains, associative algebras, Lie algebras etc. What is the ring you are after?

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Yes, this is implemented as the Vector Enumeration in magma:

This is also included in GAP.

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