# oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring

## Synopsis

• Usage:
oreIdeal(A,sigma,delta,x) or oreIdeal(A,sigma,x)
• Inputs:
• Optional inputs:
• Degree => ..., default value 1
• Outputs:

## Description

Given a ring A, an Ore extension of A by x is the quotient of the free extension A<x> by the relations x*a - sigma(a)*x-delta(a) where sigma is an automorphism of A and delta is a sigma-derivation. This method returns the defining ideal (in the appropriate tensor algebra) of an Ore extension of A by x. The current version assumes the sigma-derivation delta is 0, but this is to be fixed soon.

 i1 : B = skewPolynomialRing(QQ,(-1)_QQ,{x,y,z,w}) o1 = B o1 : FreeAlgebraQuotient i2 : sigma = map(B,B,{y,z,w,x}) o2 = map (B, B, {y, z, w, x}) o2 : RingMap B <--- B i3 : C = oreIdeal(B,sigma,a) o3 = ideal (x*y + y*x, x*z + z*x, y*z + z*y, x*w + w*x, y*w + w*y, z*w + w*z, ------------------------------------------------------------------------ - y*a + a*x, - z*a + a*y, - w*a + a*z, - x*a + a*w) o3 : Ideal of QQ <|x, y, z, w, a|>