03 - Direct Sum Decomposition of Quotients

Direct Sum Decomposition of Quotients

Let $R$ be a ring
Let $I,J$ be ideals of $R$ such that $I+J=R$ then

$$R/(I\cap J)\cong R/I\oplus R/J$$

Proof

Consider quotient maps

$$\begin{array}{rl}{q}_1& :R\to R/I\\ q_2& :R\to R/J\end{array}$$

Define

$$q:R\to R/I\oplus R/J$$

by

$$q(r)=(q_1(r),q_2(r))$$

By First Isomorphism Theorem then result follows

Showing Subjectivity
Suppose $(r+I,s+J)\in R/I\oplus R/J$ then
Since $R=I+J$ then

$$r=i_1+j_2\text{ and }s=i_2+j_2\text{ where }{i}_1,i_2\in I\text{ and }{j}_1,j_2\in J$$

But

$$r+I=j_1+I\text{ and }s+J=i_2+J$$

So

$$q(j_1+j_2)=(r+I,s+J)$$

Showing $\ker (q)=I\cap J$
if $q(r)=0$ then $q_1(r)=0$ and $q_2(r)=0$ so

$$r\in I\text{ and }r\in J⟹r\in I\cap J$$