Noncommutative computational package I am wondering if there is a program which can do simple operations over noncommutative rings, like expand products and substitute one expression for another.
To clarify, consider the following situation.  I have two reductions $ab\mapsto 1$ and $ca\mapsto c-1$.  If I consider the monomial $cab$ I can reduce it in two ways: $cab=c(ab)=c$ or $cab=(ca)b=(c-1)b=cb-b$.  I can combine these computations to arrive at a third reduction $cb\mapsto b+c$.
I'm in a situation where I have upwards of twelve reduction rules, and it gets very complicated doing the reductions.  I find myself making small errors.  Thus, the need for a machine to do these computations for me.
To make this more precise, is there a program where I can first input a number of reductions, and then second have it work on a monomial and spit out a reduced form?
 A: You may be satisfied by some noncommutative Gröbner basis programs: I know of the standalone Bergman and the GAP package GBNP.
A: There is this non-commutative algebra package for Mathematica that is quite extensive


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*http://www.math.ucsd.edu/~ncalg/
It can handle the symbolic computations in the question, among many other things.
A: Magma can certainly deal with that. Not sure about other packages; Singular has been approaching a non-commutative extension for years, but I'm not sure of its status.
A: You can do quick computations with http://servus.math.su.se/bergman/demo.html
(The page deals with homogeneous relations, but you can add a new variable $t$, homogenise the relations you want using it, and the add relations saying that $t$ commutes with $a$, $b$ and $c$. Doing this with your relations gives what seems to be infinitely many elements in the (lexicographic) Groebner basis)
A: You could try SAGE.  They have an online version (free) where everything is in python and you can run programs from the cloud.  The downloadable version has a bit more functionality but is harder to use.
