Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 29317. An ideal of a commutative ring R is said to be finitely generated if thereexist elements a 1 , . . . , a n in R such that every element r ∈ R can be writtenas a 1 r 1 + · · · + a n r n for some r 1 , . . . , r n in R. Prove that R satisfies theascending chain condition if and only if every ideal of R is finitely generated.18. Let D be an integral domain with a descending chain of ideals I 1 ⊃ I 2 ⊃ · · · .Show that there exists an N such that I k = I N for all k ≥ N. A ring satisfyingthis condition is said to satisfy the descending chain condition, orDCC. Rings satisfying the DCC are called Artinian rings, after EmilArtin.19. Let R be a commutative ring with identity. We define a multiplicativesubset of R to be a subset S such that 1 ∈ S and ab ∈ S if a, b ∈ S.(a) Define a relation ∼ on R × S by (a, s) ∼ (a ′ , s ′ ) if there exists an s ∈ Ssuch that s(s ′ a − sa ′ ) = 0. Show that ∼ is an equivalence relation onR × S.(b) Let a/s denote the equivalence class of (a, s) ∈ R × S and let S −1 R bethe set of all equivalence classes with respect to ∼. Define the operationsof addition and multiplication on S −1 R byas + b ta bs tat + bs=st= abst ,respectively. Prove that these operations are well-defined on S −1 R andthat S −1 R is a ring with identity under these operations. The ringS −1 R is called the ring of quotients of R with respect to S.(c) Show that the map ψ : R → S −1 R defined by ψ(a) = a/1 is a ringhomomorphism.(d) If R has no zero divisors and 0 /∈ S, show that ψ is one-to-one.(e) Prove that P is a prime ideal of R if and only if S = R \ P is amultiplicative subset of R.(f) If P is a prime ideal of R and S = R\P , show that the ring of quotientsS −1 R has a unique maximal ideal. Any ring that has a unique maximalideal is called a local ring.References and Suggested Readings[1] Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra.Addison-Wesley, Reading, MA, 1969.[2] Zariski, O. and Samuel, P. Commutative Algebra, vols. I and II. Springer-Verlag, New York, 1986, 1991.