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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:
- \( J/I \) is an ideal of \( R / I \)
[\[ R / J \cong \quad ? \]]
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).
