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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≅?]

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).