Let I, J two right ideals of a ring R such that I+J =R. Show thath the direct sum of I and J is isomorphic to the direct sum of R and the intersection of I and J.
Can anyone please give me at least a ring morphism to prove this via first isomorphism theorem or there is another way to prove or attack this problem. Thanks
There is an s.e.s. $I\cap J\to I\oplus J\to I+J$ given by $a\mapsto(a,-a)$ and $(x,y)\mapsto x+y$.
We want to show this splits, i.e. find a map $I+J\to I\oplus J$ such that $I+J\to I\oplus J\to I+J$ is the identity. Write $1=i+j$ and identify $I+J$ with $R$. The map $R\to I\oplus J$ is $r\mapsto (ir,jr)$.
Thus, the right $R$-module isomorphism $(I\cap J)\oplus R\mapsto I\oplus J$ is $(a,r)\mapsto (a+ir,-a+jr)$.