# Space of homomorphisms between two rings¶

sage.rings.homset.RingHomset(R, S, category=None)

Construct a space of homomorphisms between the rings `R` and `S`.

For more on homsets, see `Hom()`.

EXAMPLES:

```sage: Hom(ZZ, QQ) # indirect doctest
Set of Homomorphisms from Integer Ring to Rational Field
```
class sage.rings.homset.RingHomset_generic(R, S, category=None)

A generic space of homomorphisms between two rings.

EXAMPLES:

```sage: Hom(ZZ, QQ)
Set of Homomorphisms from Integer Ring to Rational Field
sage: QQ.Hom(ZZ)
Set of Homomorphisms from Rational Field to Integer Ring
```
Element
has_coerce_map_from(x)

The default for coercion maps between ring homomorphism spaces is very restrictive (until more implementation work is done).

Currently this checks if the domains and the codomains are equal.

EXAMPLES:

```sage: H = Hom(ZZ, QQ)
sage: H2 = Hom(QQ, ZZ)
sage: H.has_coerce_map_from(H2)
False
```
natural_map()

Returns the natural map from the domain to the codomain.

The natural map is the coercion map from the domain ring to the codomain ring.

EXAMPLES:

```sage: H = Hom(ZZ, QQ)
sage: H.natural_map()
Natural morphism:
From: Integer Ring
To:   Rational Field
```
zero()

Return the zero element of this homset.

EXAMPLES:

Since a ring homomorphism maps 1 to 1, there can only be a zero morphism when mapping to the trivial ring:

```sage: Hom(ZZ, Zmod(1)).zero()
Ring morphism:
From: Integer Ring
To:   Ring of integers modulo 1
Defn: 1 |--> 0
sage: Hom(ZZ, Zmod(2)).zero()
Traceback (most recent call last):
...
ValueError: homset has no zero element
```
class sage.rings.homset.RingHomset_quo_ring(R, S, category=None)

Space of ring homomorphisms where the domain is a (formal) quotient ring.

EXAMPLES:

```sage: R.<x,y> = PolynomialRing(QQ, 2)
sage: S.<a,b> = R.quotient(x^2 + y^2)
sage: phi = S.hom([b,a]); phi
Ring endomorphism of Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2)
Defn: a |--> b
b |--> a
sage: phi(a)
b
sage: phi(b)
a
```
Element
sage.rings.homset.is_RingHomset(H)

Return `True` if `H` is a space of homomorphisms between two rings.

EXAMPLES:

```sage: from sage.rings.homset import is_RingHomset as is_RH
sage: is_RH(Hom(ZZ, QQ))
True
sage: is_RH(ZZ)
False
sage: is_RH(Hom(RR, CC))
True
sage: is_RH(Hom(FreeModule(ZZ,1), FreeModule(QQ,1)))
False
```