Let R be a commutative ring and I and J be two ideals of R, how to show that IJ= the intersection of I and J if R=I+J?
A problem about ideal
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ring-theory
1 Answers
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HINT $\ $ One direction is trivial. The other direction requires further hypotheses, e.g. if $\rm\ I + J = 1\ $ then start from the fact that $\rm\ (I+J)\ \ (I\cap J) \ \subseteq\ I\:J\ $ follows by applying the distributive law. It's an ideal form of the integer law $\rm gcd(i,j)\ lcm(i,j) = i\:j\ \ $ so $\rm\ \ gcd(i,j)=1\ \Rightarrow\ lcm(i,j) = i\:j\:.$