Answer to Question #139041 in Abstract Algebra for Sohan kumar

Question

Let R be a commutative Ring with identity.let I and J be ideals of R such that I+J=Show that IJ= intersection of I and J.

Answer 1

Let $R$ be a commutative ring with identity and $I$ and $J$ be ideals of $R$ such that $I+J=R$. Let us prove that $IJ=I\cap J$.

Since $I$ is an ideal, $IJ\subset IR\subset I$. Since $J$ is an ideal, $IJ\subset RJ\subset J$. Therefore, $IJ\subset I\cap J$.

On the other hand, taking into accounr that $R$ is a commutative ring with identity, we conclude that

$I\cap J\subset (I\cap J)R\subset (I\cap J)(I+J)= (I\cap J)I+ (I\cap J)J\subset JI+IJ=IJ+IJ=IJ$.

Consequently, $IJ=I\cap J$.