Question #139041
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.
1
Expert's answer
2020-10-20T17:50:39-0400

Let RR be a commutative ring with identity and II and JJ be ideals of RR such that I+J=RI+J=R. Let us prove that IJ=IJIJ=I\cap J.


Since II is an ideal, IJIRIIJ\subset IR\subset I. Since JJ is an ideal, IJRJJIJ\subset RJ\subset J. Therefore, IJIJIJ\subset I\cap J.


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


IJ(IJ)R(IJ)(I+J)=(IJ)I+(IJ)JJI+IJ=IJ+IJ=IJI\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=IJIJ=I\cap J.








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