[Solved] Homework Problem Set #2

Let R be a ring with identity 1 6= 0.

7.4.19 Let R be a finite commutative ring with identity. Prove that every prime ideal of R is a maximal ideal.

7.5.3 Let F be a field. Prove that F contains a unique smallest subfield F₀ and that F₀ is isomorphic to either Q or Z/pZ for some prime p (F₀ is called the prime subfield of F).

[See Exercise 26, Section 3.]

8.2.2 Prove that any two nonzero elements of a P.I.D. have a least common multiple (cf. Exercise 11, Section 1).

8.2.3 Prove that a quotient of a P.I.D. by a prime ideal is again a P.I.D.

9.1.4 Prove that the ideals (x) and (x,y) are prime ideals in Q[x,y], but only the latter ideal is a maximal ideal.