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Math and science::Algebra::Aluffi

Rings. Third isomorphism theorem, an example.

Let \( a, b \in R \) be elements in a commutative ring \( R \). Use \( ( \bar{b} ) \) to denote the equivalence class of \( b \) in \( R/(a) \). Then firstly we have:

[\[ (\bar{b}) = \frac{?}{(a)} \]]
<p>and with that result it can be seen that:</p>
[ \[ \frac{R/(a)}{(\bar{b})} \cong \quad ? \]]

Recap of ring generators:

Ring generators

Let \( a \in R \) be an element of a ring. The subset \( Ra \) is a left-ideal and the subset \( aR \) is a right-ideal. If \( R \) is a commutative ring, the ideals coincide, and the syntax \( (a) \) is used to denote the ideal.