Second Isomorphism Theorem (Ring)

definitionring-theory

Definition

For ideals $I,J⊴R$, then we have an isomorphism of rings (without unity):

$${}^{(I+J)}{/}_J\simeq {}^I{/}_{I\cap J}.$$

An isomorphism is $F:{}^I{/}_{I\cap J}\to {}^{I+J}{/}_J$ given by $F(x+I\cap J)=x+J$.

Explanation of Terms

$J⊴R$ $J\subseteq I+J⊴R$ is an ideal, so it is a subring (maybe without unity), therefore $J⊴I+J$ is an ideal, so ${}^{I+J}{/}_J$ is a ring (maybe without unity).

Similarly, $I\cap J\subseteq I⊴R⟹{}^I{/}_{I\cap J}$ is a ring (maybe without unity).

Examples