Quotients and direct products

Let R1,,Rk be commutative rings, and set R=R1××Rk.

  1. Let IjRj be ideals, and put I=I1××Ik. Use the First Isomorphism Theorem to prove that R/IR1/I1××Rk/Ik.
  2. Prove the prime ideals of R have the form R1××Rj1×Pj×Rj+1××Rk where PjRj is a prime ideal for 1jk. (Omit the proof that this is an ideal.)