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 Image

apply ring homomorphism
 Syntax
 ``` Image(R::E:OBJECT, F:TAGGED("RMap")):OBJECT Image(V:OBJECT, F:TAGGED("RMap")):OBJECT where R is the identifier for a ring and F has the form RMap(F_1:POLY,...,F_n:POLY) or the form RMap([F_1:POLY,...,F_n:POLY]). The number n is the number of indeterminates of the ring R. In the second form, V is a variable containing a CoCoA object dependent on R or not dependent on any ring. ```

 Description
This function maps the object E from one ring into the current ring as determined by F. Suppose the current ring is S, and E is an object dependent on a ring R; then
```               Image(R::E, F)
```
returns the object in S obtained by substituting F_i for the i-th indeterminate of R in E. Effectively, we get the image of E under the ring homomorphism,
```               F: R   --->  S
x_i |--> F_i,
```
where x_i denotes the i-th indeterminate of R.

Notes:

1. The coefficient rings for the domain and codomain must be the same.

2. If R = S, one may use Image(E, F) but in this case it may be easier to use Eval or Subst .

3. The exact domain is never specified by the mapping F. It is only necessary that the domain have the same number of indeterminates as F has components. Thus, we are abusing terminology somewhat in calling F a map.

4. The second form of the function does not require the prefix R:: since the prefix is associated automatically.

5. If the object E in R is a polynomial or rational function (or list, matrix, or vector of these) which involves only indeterminates that are already in S, the object E can be mapped over to S without change using the command BringIn.

 Example
 ``` Use C ::= QQ[u,v]; -- domain Use B ::= QQ[x,y]; -- another possible domain I := Ideal(x^2-y); -- an ideal in B Use A ::= QQ[a,b,c]; -- codomain F := RMap(a,c^2-ab); Image(B::xy, F); -- the image of xy under F:B --> A -a^2b + ac^2 ------------------------------- Image(C::uv, F); -- the image of uv under F:C --> A -a^2b + ac^2 ------------------------------- Image(I, F); -- the image of the ideal I under F: B --> A Ideal(a^2 + ab - c^2) ------------------------------- I; -- the prefix "B::" was not needed in the previous example since -- I is already labeled by B B :: Ideal(x^2 - y) ------------------------------- Image(B::Module([x+y,xy^2],[x,y]), F); -- the image of a module Module([-ab + c^2 + a, a^3b^2 - 2a^2bc^2 + ac^4], [a, -ab + c^2]) ------------------------------- X := C:: u+v; -- X is a variable in the current ring (the codomain), A, X; -- whose value is an expression in the ring C. C :: u + v ------------------------------- Image(X, F); -- map X to get a value in C -ab + c^2 + a ------------------------------- ```