Correspondence Theorem for Quotients

theoremring-theory

Formal Definition

Given commutative ring $(R,+,\cdot )$ with $1_R$, and ideal $I⊴R$, there exists a morphism:

$$\phi :R\to S={}^R{/}_I,\phi (r)=r+I.$$

where the $\ker \phi =I$, and $\phi$ is a surjective morphism.

There are bijective correspondences between:

$$\{J⊴R\mid I\subseteq J\}\text{, and }\{a⊴S={}^R{/}_I\}$$

Proof

We already know from groups, $({}^J{/}_I,+)\le ({}^R{/}_I,+)$.

If $x\in J,x+I\in {}^J{/}_I$, and if $r\in R,r+I\in {}^R{/}_I$. Thus, $(r+I)(x+I)=rx+I\in {}^J{/}_I$.

Then, the proof is as for groups.

Examples