# map(Module,Module,RingElement) -- construct the map induced by multiplication by a ring element on the generators

## Synopsis

• Function: map
• Usage:
map(M,N,r)
• Inputs:
• M,
• N, , over the same ring R as M. An integer here stands for the free module of that rank.
• r, , in the ring R
• Optional inputs:
• Degree => ..., default value null, specify the degree of a map
• DegreeLift => ..., default value null, make a ring map
• DegreeMap => ..., default value null, make a ring map
• Outputs:
• , The map induced by multiplication by r on the generators

## Description

If r is not zero, then either M and N should be equal, or they should have the same number of generators. This gives the same map as r * map(M,N,1). map(M,N,1) is the map induced by the identity on the generators of M and N.
 i1 : R = QQ[x]; i2 : map(R^2,R^3,0) o2 = 0 2 3 o2 : Matrix R <--- R i3 : f = map(R^2,R^2,x) o3 = | x 0 | | 0 x | 2 2 o3 : Matrix R <--- R i4 : f == x *map(R^2,R^2,1) o4 = true

## Caveat

If M or N is not free, then we don't check that the the result is a well defined homomorphism.