Ring homomorphisms and isomorphisms

Just as in Group theory we look at maps which "preserve the operation", in Ring theory we look at maps which preserve both operations.

Definition

A map f : R→ S between rings is called a ring homomorphism if
f(x + y) = f(x) + f(y) and f(xy) + f(x)f(y) for all x, y ∈ R.

Remarks

1. The operations on the left are in the ring R; those on the right are in S.

2. Since such a homomorphism is a group homomorphism from (R, +) to (S, +) it maps 0_R to 0_S.
   Even if the rings R and S have multiplicative identities a ring homomorphism will not necessarily map 1_R to 1_S.

3. It is easy to check that the composition of ring homomorphisms is a ring homomorphism.
   

A ring homomorphism which is a bijection (one-one and onto) is called a ring isomorphism.

If f : R → S is such an isomorphism, we call the rings R and S isomorphic and write R S.

Remarks

1. Isomorphic rings have all their ring-theoretic properties identical. One such ring can be regarded as "the same" as the other.
   
2. The inverse map of the bijection f is also a ring homomorphism.
   

1. The map from Z to Z_n given by x ↦ x mod n is a ring homomorphism. It is not (of course) a ring isomorphism.

2. The map from Z to Z given by x ↦ 2x is a group homomorphism on the additive groups but is not a ring homomorphism.

3. The map from Z to the ring of 2 × 2 real matrices given by x ↦ is a ring homomorphism which does not map the multiplicative identity to the multiplicative identity.

   Of course the map x ↦ x mod n is a homomorphism which does map the identity to the identity.
   

4. The "evaluation at ¹/₂ map" from the ring of continuous real-valued functions on the interval [0, 1] to R given by (e_½)f = f (¹/₂) for f belongs C[0, 1] is a ring homomorphism.

5. The "evaluation at 1 map" from the ring Q[x] to Q given by (e₁)p = p(1) for p ∈ Q[x] is a ring homomorphism.
   That is: e₁(a₀ + a₁x + a₂x² + ... + a_nxⁿ) = a₀ + a₁ + a₂ + ... + a_n.

6. The "evaluation at √2 map" from Z[x] to R given by p ↦ p(√2) is a ring homomorphism.
   This will turn out to be a surprisingly important example.

7. The map from C to ring of 2 × 2 real matrices given by a + bi ↦ is a ring isomorphism.

   Proof: Exercise.

8. Let R be the field with 9 elements {a + bx | a, b ∈ Z₃} and the multiplication rule x² = -1.
   Let S be the field with 9 elements {a + by | a, b ∈ Z₃ } and the multiplication rule y² = y + 1.
   Then the map defined by 1 ↦ 1 and x ↦ y + 1 defines a ring isomorphism.

   Proof: We'll see a neat way of proving this later.
   

Definition

The kernel of a (ring) homomorphism is the set of elements mapped to 0.

That is, if f: R→ S is a ring homomorphism, ker(f) = f^-1(0) = {r ∈ R | f(r) = 0_S }.

Theorem

The kernel of a ring homomorphism is an ideal.

Proof
An easy verification.

Remarks

1. Note the similarity with the corresponding result for groups: the kernel of a group homomorphism is a normal subgroup.

2. If the ring R is not commutative, the kernel is a two-sided ideal.
   

1. The kernel of the above map from Z to Z_n is the ideal nZ.

2. Just as in the group-theory case, the kernel of a homomorphism is {0} if and only if the homomorphism is one-one.
   Proof
   (=>) If f(r) = f(s) then f(r - s) = 0 and so r - s belongs ker(f) and we have r - s = 0.
   (<=) If a ∈ ker(f) and a ≠ 0 then a, 0 ↦ 0 and so the map is not one-one.

3. The kernel of the "evaluation at 0" map from R[x] to R is the ideal < x > of polynomials with zero constant terms.

4. The kernel of the "evaluation at 0 taken modulo 2" map from Z[x] to Z₂ is the ideal < 2, x > of polynomials with even constant terms.
   

We will see later that every ideal is the kernel of a ring homomorphism. This is similar to the group theory result that every normal subgroup is the kernel of a group homomorphism.

The last result in this section also parallels the corresponding example in Group theory,

Theorem

The image of a ring homomorphism f: R→ S is a subring of S.

Proof

The image is the set im(f) = {s ∈ S | s = f(r) for some r ∈ R }. It is easy to do the verification.

Example

The image of e_√2 from Z[x] to R is the subring {a + b √2 | a, b ∈ Z } we met earlier.