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9.3.2 Problem. Let R be a ring and I an ideal of R. Let π :...

9.3.2 Problem. Let R be a ring and I an ideal of R. Let π : R→R/I be the natural projection. Let J be an ideal of R.

  1. Show that π−1(π(J)) = (I, J).
  2. Show that if J is a maximal ideal of R with, I not ⊆ J, then π (J) = R/I.
  3. Suppose that J is an ideal of R with I ⊆ J. Show that J is a maximal ideal of R if and only if π(J) is a maximal ideal of R/I.
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