Prove the smallest ideal of r containing i and j

Question

Questions:

Ideals

Please answer each part in full:

Let I and J be ideals of R:

(a) Prove that I+J is the smallest ideal of R containing both I and J;

(b) Prove that IJ is an ideal contained in I (intersection) J

(c) Give an example where IJ is not equal to I (intersection) J

(d) Prove that if R is commutative and if I+J=R, then IJ = I (intersection) J.