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Ring Homomorphisms and Ideals

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Ring Homomorphism

A ring homomorphism is a map between rings which is compatible with addition, multiplication and maps the unit of to that of (if they are unital). A ring homomorphism which admits an inverse is an isomorphism.

Proposition

For every ring there is a unique homomorphism .

Proof We must have ; and then ( ones) for ; and . It is easy to check that this is multiplicative as well.

Characteristic

Let be the unique ring homomorphism from the integers to a ring . Then the characteristic of is the smallest nonnegative integer such that .
i.e., is the smallest positive integer such that in , or if no such integer exists.

e.g. The characteristic of is , the characteristic of is .

Proposition

.

Kernel and Image

The kernel and image of a ring homomorphism are

Remark

The image is a subring of because , , . The kernel is an ideal of , which is defined in the next section.

Ideals

Ideal

A left ideal in a ring is a subset of such that

  • ;
  • is closed under addition;
  • If then for all .

Equivalently, it is a non-empty subset such that for we have for every . We write in such case.

Ideals to Rings are Normal Subgroups to Groups

Ideals are the ring-theoretic analogue of normal subgroups in group theory. Just as normal subgroups allow us to form quotient groups, ideals allow us to form quotient rings.

Generated Ideal

Let . The set is an ideal in called the ideal generated by . The notation is .
An ideal is principal if it is generated by one element. That is, there exists such that . It is denoted .

e.g. In every ideal is principal and takes the form . (ref. proposition).

Product Ideal

For , define the product ideal as

Proposition

If are ideals so are and .

Proof It is clear that is an ideal. For , we have since . If and are in , then . If , then .

Quotient rings

Quotient Ring

Let be a ring and an ideal. Then, as an additive subgroup, is normal in and we can form the quotient with addition , zero , and inverse . There is a unique way to upgrade this abelian group to a ring, called the quotient ring. Define

Thrm Mapping Property of Quotients

Let be a ring homomorphism with kernel and let be another ideal. Let be the canonical map. Then if ,there exists a unique ring homomorphism such that .

First Isomorphism Theorem

Let be a ring homomorphism with kernel . If is surjective then is an isomorphism.

e.g. .

Correspondence Theorem

Let be a surjective ring homomorphism with kernel . There is a bijective correspondence:

Misplaced & \{\text{Ideals of } R \text{ containing } K\} &\longleftrightarrow\{\text{Ideals of } R^\prime\} \\ I &\shortmid\!\longrightarrow \varphi(I) \\ I^\prime &\longleftarrow\!\shortmid \varphi^{-1}(I^\prime) \end{aligned} $$ If the ideal $I$ corresponds to the ideal $I^\prime$ then $R/I \cong R^\prime/I^\prime$.

Corollary

Let be a ring, suppose . Then the ideals of corresponds to ideals of containing .

Chinese Remainder Theorem

Lemma

Suppose, , then .

Proof Define by . It is easy to check that is a ring homomorphism (actually, it is the product of two quotient maps). The kernel of is , and the image is all of . By the first isomorphism theorem, we have .

Chinese Remainder Theorem

Let such that , then and .

Proof We first show that . It is clear that . Conversely, for all , note that since , we have for some . Then . Thus . Therefore, combining the above lemma, we have .

Remark

Given , requires being a commutative unital ring, while holds for any ring .

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