next | previous | forward | backward | up | index | toc

# isModuleFinite(RingMap) -- whether the target of a ring map is finitely generated over source

## Synopsis

• Function: isModuleFinite
• Usage:
isModuleFinite f
isModuleFinite R
• Inputs:
• f, , or $R$ a ring
• Outputs:

## Description

A ring map $f \colon A \to B$ makes $B$ into a module over $A$. This method returns true if and only if this module is a finitely generated $A$-module.

 i1 : kk = QQ; i2 : A = kk[t]; i3 : C = kk[x,y]; i4 : B = C/(y^2-x^3); i5 : f = map(A, B, {t^2, t^3}) 2 3 o5 = map (A, B, {t , t }) o5 : RingMap A <-- B i6 : isWellDefined f o6 = true i7 : isModuleFinite f o7 = true
 i8 : f = map(kk[x,y], A, {x+y}) o8 = map (QQ[x..y], A, {x + y}) o8 : RingMap QQ[x..y] <-- A i9 : assert not isModuleFinite f

If a ring $R$ is given, this method returns true if and only if $R$ is a finitely generated module over its coefficient ring.

 i10 : A = kk[x] o10 = A o10 : PolynomialRing i11 : B = A[y]/(y^3+x*y+3) o11 = B o11 : QuotientRing i12 : isModuleFinite B o12 = true