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, b)}{(a)}

<p>and with that result it can be seen that:</p>

\frac{R / (a)}{(\bar{b})} ≅ R / (a, b)

Recap of ring generators:

Ring generators

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

The third isomorphism theorem:

Third isomorphism theorem for rings.

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

$J / I$ is an ideal of $R / I$
$R / J ≅ \frac{R / I}{J / I}$

And really, it's the general case that one should commit to intuition:

Isomorphism theorem for rings (general)

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

$\tilde{φ}$ is defined as:

\tilde{φ} (r + I) = φ (r)