# RingMap RingElement -- apply a ring map

## Synopsis

• Operator: SPACE
• Usage:
f X
• Inputs:
• f, , a ring map from R to S.
• X, , an ideal, , , , or
• Outputs:
• , the image of X under the ring map f. The result has the same type as X, except that its ring will be S.

## Description

If X is a module then it must be either free or a submodule of a free module. If X is a chain complex, then every module of X must be free or a submodule of a free module.
 i1 : R = QQ[x,y]; i2 : S = QQ[t]; i3 : f = map(S,R,{t^2,t^3}) 2 3 o3 = map (S, R, {t , t }) o3 : RingMap S <--- R i4 : f (x+y^2) 6 2 o4 = t + t o4 : S i5 : f image vars R o5 = image | t2 t3 | 1 o5 : S-module, submodule of S i6 : f ideal (x^2,y^2) 4 6 o6 = ideal (t , t ) o6 : Ideal of S i7 : f resolution coker vars R 1 2 1 o7 = S <-- S <-- S <-- 0 0 1 2 3 o7 : ChainComplex

## Caveat

If the rings R and S have different degree monoids, then the degrees of the image might need to be changed, since Macaulay2 sometimes doesn't have enough information to determine the image degrees of elements of a free module.

## Ways to use this method:

• "RingMap ChainComplex"
• "RingMap Ideal"
• "RingMap Matrix"
• "RingMap Module"
• RingMap RingElement -- apply a ring map
• "RingMap Vector"