Dear GAP Forum and Alper, You can construct quotients of polynomial rings in the following way in GAP:

... GAP, Version 4.7.5 of 24-May-2014 (free software, GPL) ... gap> R := PolynomialRing(Rationals,["a"]); Rationals[a] gap> a := IndeterminatesOfPolynomialRing(R)[1]; a gap> I := Ideal(R, [a*a + 1]); <two-sided ideal in Rationals[a], (1 generators)> gap> S := R/I; <ring Rationals,(1),(a)> Note that I am using version 4.7.5, since in some of the earlier versions, the latter command gave an error message. You can also construct non-commutative polynomial rings as follows: gap> R := FreeAlgebraWithOne(Rationals, 2); <algebra-with-one over Rationals, with 2 generators> gap> gens := GeneratorsOfAlgebra(R); [ (1)*<identity ...>, (1)*x.1, (1)*x.2 ] gap> x1 := gens[2]; (1)*x.1 gap> x2 := gens[3]; (1)*x.2 gap> I := Ideal(R, [x1^2, x1*x2 + x2*x1, x2^2]);

`<two-sided ideal in <algebra-with-one over Rationals, with 2`

`generators>, (3 generators)>`

gap> S := R/I; <algebra-with-one over Rationals, with 2 generators> A third way of constructing finite dimensional algebras is to use the undeposited package QPA, which you find at http://sourceforge.net/projects/quiverspathalg/ and see http://www.math.ntnu.no/~oyvinso/QPA/ for further information. With this package you can construct and do

`computations for finite dimensional quotients of path algebras. In`

`particular`

construct modules/representations, homomorphisms over these and do some homological algebra over these algebras. gap> Q := Quiver(1, [[1,1,"a"], [1,1,"b"]]); <quiver with 1 vertices and 2 arrows> gap> KQ := PathAlgebra(Rationals, Q); <Rationals[<quiver with 1 vertices and 2 arrows>]> gap> AssignGeneratorVariables(KQ); #I Assigned the global variables [ v1, a, b ] gap> relations := [ a*a, a*b + 2*b*a, b*b]; [ (1)*a^2, (1)*a*b+(2)*b*a, (1)*b^2 ] gap> A := KQ/relations; # the quantum plane <Rationals[<quiver with 1 vertices and 2 arrows>]/ <two-sided ideal in <Rationals[<quiver with 1 vertices and 2 arrows>]>, (3 generators)>> Best regards, Oeyvind Solberg. _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum