Let R be an integral domain (recall this means R is commutative with a non-zero identity, and no zero-divisors). (a) Prove that if every ascending chain of principal ideals (:r:)sube(:r_(1):)sube(:r_(2):)subecdots stabilizes (that is, there is some NinZ so that (:r_(i):)=(:r_(N):) for every i>=N ), then r has a factorization into irreducibles. Recall: r has a factorization into irreducibles means r=r_(1)cdotsr_(k) for some kinZ where r_(1),dots,r_(k) are irreducible. Hint: Assume r does not have a factorization into irreducibles and produce an infinite ascending chain as above that does not stabilize. (b) Show that if R is Noetherian (see last homework!), every element of R has a factorization into irreducibles.