Starting in polynomial extents, computational commutative algebra, branched into applications via Gröbner basis theory and the generalized concept of approximate commutative algebra. But what is next? Do you believe the computation extends to non-commutative algebra?
It seems that non-commutative Groebner bases have been for long in relation with differential algebra.
Of course. Since many years there are various flavours of noncommutative Groebner bases around. There are books as wells as computer algebra systems. It's a different question - what kind of noncommutativity one wants, for there are many very different possibilities and needs.
@Patrick: yes, many of them. I am practicing non-commutative computer algebraist and developed many projects since 1998 in this field.
(1) In a paper "Letterplace Groebner bases ..." with R. La Scala you will find an overview of computer algebra approaches to free associative algebras, with references to computer algebra systems (Bergman, MAGMA, Singular:Letterplace, GBNP@GAP, NCALGEBRA@MATHEMATICA and so on). (2) As for Noetherian non-commutative algebras with PBW basis (like G-algebras = algebras of solvable type = PBW algebras), I studied them from the point of view of computer algebra in my Ph.D. thesis (2005). There are links and descriptions of systems. In the framework of my Ph.D. I've developed the system Singular:Plural, being one of the fastest systems, doing Groebner bases over PBW algebras. And I can continue...
Quite early algorithmic work has been done by Kandri-Rody and Weispfenning:
A. Kandri-Rody and V. Weispfenning. Non-commutative Gröbner bases in algebras of solvable type. Journal of Symbolic Computation, 9(1):1–26, 1990.
With respect to the original question, research in non-commutative algebra is hardly a next step in commutative algebra.
@Thomas: a detailed historical overview can be found in my dissertation, and yet more expanded in the paper "Letterplace Groebner bases ..." by R. La Scala and myself. Actually the algorithmic story in this realm started much earlier than 1990, see e.g. the papers by Teo Mora from 1988 and Apel&Lassner from 1988.
We have developed the Gröbner theory of ideals and modules for skew PBW
Extensions; skew PEN extensions is an alternative way of interpreting many non-commutative rings and quantum algebras.
Some questions can be solved in a non-commutative graded case:
http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=fpm&paperid=654&option_lang=eng
I will spend time in finding the dimension without completely evaluating the reduced grl-Grobner basis. This idea is still in its infancy. So, until further development no more comments can be added.
@Rostam, you need just a Groebner basis with respect to a positive weighted degree ordering for reading off the Krull dimension, "reduced" or "grl" are not necessary. Your Groebner basis also need not be minimal (i.e. you can have elements which leading monomials are divisible by the others). In such a case you extract leading monomials and perform an easy interreduction on this set of monomials. The Krull dimension is determined by the ideal of leading monomials, therefore you only need to know that your starting set is a Groebner basis.
I would rather not to reveal all the details until further clarifications. I started off my experiments with graded rev. lex order which is quite adequate for initial experimentations. Later the general setting of the theory certainly can be developed. Certainly, a trace of leading monomials is the target when a stopping criterion is established to announce the Krull dimension. The starting set would never be a Grobner basis I suppose!!!! You may get a stopping criterion.
I have done many experiments last year. Then I left it for some time. If you want to follow it, be advised that some submodules of the syzygy modules of S-polynomials will be generated. The stopping criterion must discard this generation of extra syzygies. To me it was a kind of difficult to establish a complete set of proved facts. Probably later I start over.
@Rostam: this sounds interesting, go on! However, consider the following situation: in the set of polynomials you start with, certain powers of all variables are present. Thus the Krull dimension is either 0 (if the ideal does not contain 1) or -\infty otherwise. This shows, that without computing Groebner basis you can only give estimates from above...
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