7. The First Isomorphism Theorem has two important corollaries: the Second Isomorphism Theorem andthe Third Isomorphism Theorem. For this exam, we will investigate the Third Isomorphism Theoremfor rings: Theorem 0.1.

(Third Isomorphism Theorem for rings) Let I and J be ideals of ring R, with I C J.Then I is an ideal of J, and (R / I) /(J / I) \simeq R / J Note that this theorem allows us to greatly simplify cases where we would construct a factor ring out ofanother factor ring. For example, (Z/18Z)/(9Z/18Z) = Z/9Z This question will walk you through the steps for the proof of the Third Isomorphism Theorem. Considerthe homomorphism o : R/I → ŘĮJ with 6(a+I) = a+J. (I will allow you to assume o is a well-definedhomomorphism on this exam). Prove I is an ideal of J. (This allows us to define J/I) \text { Prove that ker } \phi=J / I \text {. } \text { s) Prove } \phi(R / I)=R / J \text {, i.e. prove } \phi \text { is onto. } d) (4 points) Use the First Isomorphism Theorem on o to prove the Third Isomorphism Theorem.(Using parts (a),(b), and (c), you are able to write part (d) with only one or two lines of proof).