# map(Ring,Ring,List) -- make a ring map

## Synopsis

• Function: map
• Usage:
map(R,S,m)
• Inputs:
• R, a ring, the target ring
• S, a ring, the source ring
• m, a list, of n elements of R, where n is the number of variables in the polynomial ring S, or a list of pairs x => r, where x is a generator of S and r is an element of R, specifying that x is to be sent to 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 ring homomorphism from S to R which sends the i-th variable of S to the i-th entry in m, or in the second case, performs the indicated substitutions.

## Description

 i1 : R = ZZ[x,y]; i2 : S = ZZ[a,b,c]; i3 : f = map(R,S,{x^2,x*y,y^2}) 2 2 o3 = map (R, S, {x , x*y, y }) o3 : RingMap R <-- S i4 : f(a+b+c^2) 4 2 o4 = y + x + x*y o4 : R i5 : g = map(R,S,{a=>x^2,b=>x*y,c=>y^2}) 2 2 o5 = map (R, S, {x , x*y, y }) o5 : RingMap R <-- S i6 : g(a+b+c^2) 4 2 o6 = y + x + x*y o6 : R