UNIVERSITY OF CAMBRIDGE MATHEMATICAL TRIPOS Part III Monday, 1 June, 2015 9:00 am to 12:00 pm PAPER 1 COMMUTATIVE ALGEBRA Attempt no more than FOUR questions. There are SIX questions in total. The questions carry equal weight. Except in question 6, all rings are assumed to be commutative. STATIONERY REQUIREMENTS Cover sheet Treasury Tag Script paper SPECIAL REQUIREMENTS None You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
UNIVERSITY OF CAMBRIDGE 2 1 Define what it means for a ring R to be Noetherian. Let R be a Noetherian ring and let f be a ring homomorphism from R to R. Show that if f is surjective than it is also injective. Define the nilradical N (R) of a ring R. Show that it is equal to the intersection of all the prime ideals of R, and, if R is Noetherian, that it is the intersection of finitely many prime ideals. Show, for any ring R, that N (R) = 0 if and only if N (R[X]) = 0. Give an example of a ring whose nilradical is not nilpotent. 2 (i) What does it mean for an ideal of a ring R to be prime? Let P1, P2 and P3 be prime ideals of R with Pi ? Pj only if i = j. Show that their union U is not an ideal. Give an example to show that if the ideals Pi are not necessarily prime then their union may be an ideal. (ii) What does it mean for a subset S of R to be multiplicatively closed? Let S be a multiplicatively closed subset of R. Show that the set of prime ideals of the localisation S-1R corresponds bijectively to the set of prime ideals of R disjoint to S. (iii) A multiplicatively closed subset S is said to be saturated if xy I S only if both x I S and y I S. Show that a multiplicatively closed subset of R is saturated if and only if its complement in R is a union of prime ideals. Give an example of a ring R and a saturated multiplicatively closed subset S of R whose complement in R is not a union of finitely many prime ideals. 3 (i) What does it mean for a ring T to be integral over a subring R? Suppose T is integral over R. Let P be a prime ideal of R. Show that there is a prime ideal Q of T with Q n R = P. (ii) Let k be a field and let I be an ideal of the Laurent polynomial algebra A= k[X1, X 1 , X2, X21, X3, X31 ]. Let T = A/I. Show that if T is not finite dimensional as a k-vector space then it is integral over some subring isomorphic to a Laurent polynomial algebra k[Y1, Y11, ... , Ym, Ym1] for some m with 1 < m <