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Math and science::Algebra::Aluffi

Rings. Third isomorphism theorem.

This theorem has the feel of preserving a quotient through the double mapping to a shared quotient.

Third isomorphism theorem for rings

Let \( R \) be a ring. Let \( I \) and \( J \) be ideals of \( R \), with \( I \) being a "smaller" ideal contained within \( J \). Then:

  1. \( J/I \) is an ideal of \( R / I \)
  2. \[ R / J \cong \frac{R/I}{J/I} \]

Can you remember the proof?

The proof of the theorem uses (twice!) the general isomorphism theorem for rings (Aluffi doesn't give it a name, it's just Theorem 3.8 in his book).


Isomorphism theorem for rings (general)

Let \( R \) and \( S \) be a rings, and let \( \varphi : R \to S \) be a ring homomorphism. \( \varphi \) induces an ideal \( \operatorname{ker}(\varphi) \). Consider two-sided ideals contained in \( \operatorname{ker}(\varphi) \). Let \( I \subseteq \operatorname{ker}(\varphi) \) be one of these ideals. Then there is a ring homomorphism \( \widetilde{\varphi} \) fully determined by \( \varphi \) and \( I \) such that the following diagram commutes:

\( \widetilde{\varphi} \) is defined as:

\[ \widetilde{\varphi}(r + I) = \varphi(r) \]

Third isomorphism theorem. Proof idea.

\( J \) is the larger ideal and induces a smaller quotient ring \( R/J \) through a homomorphism we will label as \( \varphi \). \( J \) is the kernel of this homomorphism. \( I \) is a subset of \( J \) and is also sent to \( 0_{R/J} \) by \( \varphi \). If we consider \( R/I \), we have a homomorphism \( \widetilde{\varphi} \) that is fully determined by \( \varphi \) and \( I \) that sends (many-to-one) cosets in \( R/I \) to elements in \( R/J \). It is the general isomorphism theorem that tells us this. We apply this process again to break the surjective \( \widetilde{\varphi} \) into an isomorphism! We do this by identifying the kernel of the homomorphism, \( \operatorname{ker}\widetilde{\varphi} \). It is the set of cosets in \( R/I \) that are sent to \( 0_{R/J} \). These are isomorphic to \( J/I \).

Example

\( 10\mathbb{Z} \) induces a quotient ring \( \mathbb{Z}/10\mathbb{Z} \) with 10 elements. \( 30\mathbb{Z} \) is a subset of \( 10\mathbb{Z} \) and induces a quotient ring \( \mathbb{Z}/30\mathbb{Z} \) with 30 elements. The three elements in \( \mathbb{Z}/30\mathbb{Z} \) that are sent to \( 0_{\mathbb{Z}/10\mathbb{Z}} \) are \( (0, 10, 30 ) \) and they are isomorphic to \( 10\mathbb{Z}/30\mathbb{Z} \).

First isomorphism theorem

In the proof above, the second application of the general isomorphism theorem is usually called the first isomorphism theorem for rings. It is recalled here:

First isomorphism theorem for rings

Let \( R \) and \( S \) be rings. Let \( \phi: R \to S \) be a surjective ring homomorphism. Then:

\[ \frac{R}{\operatorname{ker}(\phi)} \sim S \]

A linear projection?

The theorem suggests a linear operation feel—that of projecting into the \( R/I \) ideal:

\[ \operatorname{proj}_{J/I}(\operatorname{ideal}(R, J)) = \operatorname{ideal}(\operatorname{proj}_{J/I}(R), \operatorname{proj}_{J/I}(J)) \]

Source

Aluffi p141, p142