# presentation(PolynomialRing,QuotientRing) -- presentation of a quotient ring

## Synopsis

• Function: presentation
• Usage:
presentation B
presentation(A,B)
• Inputs:
• A,
• B, , a quotient ring of A
• Outputs:
• , whose image is the ideal of A defining B

## Description

If A is not present, then it is understood to be the ultimate ambient polynomial ring of B. In general, A may be any ring of which B is a quotient.

In the examples below, A is the ultimate ambient polynomial ring of A, B and C.

 i1 : A = QQ[a..d]; i2 : B = A/(a^2,b^3); i3 : C = B/(a*b*c,b*c*d, b^2); i4 : presentation A o4 = 0 1 o4 : Matrix A <--- 0 i5 : presentation B o5 = | a2 b3 | 1 2 o5 : Matrix A <--- A i6 : presentation C o6 = | abc bcd b2 a2 b3 | 1 5 o6 : Matrix A <--- A i7 : presentation(B,C) o7 = | abc bcd b2 | 1 3 o7 : Matrix B <--- B i8 : presentation(A,C) o8 = | abc bcd b2 a2 b3 | 1 5 o8 : Matrix A <--- A i9 : minimalPresentation C QQ[a..d] o9 = -------------------------- 2 3 2 (a , b , a*b*c, b*c*d, b ) o9 : QuotientRing

## Caveat

The given presentation is often not minimal