# CoCoA:Image

## Image

apply ring homomorphism

### 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 <verbatim>

```              Image(R::E,F)
```

</verbatim> 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, <verbatim>

```              F: R   --->  S
x_i |--> F_i,
```

</verbatim> 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 <ttref>Eval</ttref> or <ttref>Subst</ttref>.
```

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 ::= Q[u,v];   -- domain
Use B ::= Q[x,y];   -- another possible domain
I := Ideal(x^2-y);  -- an ideal in B
Use A ::= Q[a,b,c]; -- codomain
F := RMap(a,c^2-ab);
Image(B::xy, F);    -- the image of xy under F:B --&gt; A
-a^2b + ac^2
-------------------------------
Image(C::uv,F);     -- the image of uv under F:C --&gt; A
-a^2b + ac^2
-------------------------------
Image(I,F);         -- the image of the ideal I under F: B --&gt; A
Ideal(a^2 + ab - c^2)
-------------------------------
I; -- the prefix <quotes>B::</quotes> 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
-------------------------------
```

### Syntax

```Image(R::E:OBJECT,F:TAGGED(<quotes>RMap</quotes>)):OBJECT
Image(V:OBJECT,F:TAGGED(<quotes>RMap</quotes>)):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.
```

```   <type>ring</type>
```