# 39.5 Operators for Finitely Presented Algebras

`A / relators`

returns a finitely presented algebra that is the quotient of the free algebra A (see FreeAlgebra) by the two-sided ideal in A spanned by the elements in the list relators.

This is the general method to construct finitely presented algebras in GAP. For the special case of group algebras of finitely presented groups see FpAlgebra.

`A ^ n`

returns a free A-module of dimension n (see chapter Modules) for the finitely presented algebra A.

```    gap> f:= FreeAlgebra( Rationals, 2 );
UnitalAlgebra( Rationals, [ a.1, a.2 ] )
gap> a:= f / [ f.1^2 - f.one, f.2^2 - f.one, (f.1*f.2)^2 - f.one ];
UnitalAlgebra( Rationals, [ a.1, a.2 ] )
gap> a = f;
false
gap> a^2;
Module( UnitalAlgebra( Rationals, [ a.1, a.2 ] ),
[ [ a.one, a.zero ], [ a.zero, a.one ] ] ) ```

`a in A`

returns `true` if a is an element of the finitely presented algebra A, and `false` otherwise. Note that the answer may require the computation of an isomorphic matrix algebra if A is not a parent algebra.

```    gap> a.1 in a;
true
gap> f.1 in a;
false
gap> 1 in a;
false ```

GAP 3.4.4
April 1997