Math 614, Fall 2012
Problem Set #1
Due: Wednesday, September 26
1. Let R be a commutative ring and x an indeterminate over R.
(a) Show that an element of R[x] is nilpotent if and only if all of its coecients are nilpotent.
(b) Characterize the elements of
Math 614, Fall 2012
Due: Friday, October 12
Problem Set #2
1. (a) Let X = Spec (R) and let cfw_D(f ) : be a family of open sets such that
any nite number of elements in the family have nonempty intersection. Show that the
intersection of the family is no
Math 614, Fall 2012
Due: Monday, October 29
Problem Set #3
1. Let R = K[x1 , . . . , xn ] be the polynomial ring in n variables over a eld K. Let
k be an integer with 0 k < n and let Ik be the ideal generated by all of the products
xi xi+1 xi+k , where th
Math 614, Fall 2012
Problem Set #3: Solutions
1. Any prime will have to contain at least one variable from each monomial in Ik . The
minimal primes are therefore generated by subsets of the variables minimal with respect
to the property that they contain
Math 614, Fall 2012
Problem Set #1: Solutions
1. (a) The sum or dierence of two nilpotents is nilpotent: if am = 0 and bn = 0, then
each term in (a b)m+n1 involves neither am or bn . It is then clear that a polynomial
is nilpotent if all coecients are. To
Math 614, Fall 2012
Problem Set #2: Solutions
1. (a) Let W be the multiplicative system whose elements are 1 and all nite products of
the f . If 0 W there is a prime P disjoint from W , and P is clearly in the intersection.
/
n
But if 0 W , then some prod
Math 614, Fall 2012
Due: Friday, December 7
Problem Set #5
1. Let R be a ring, and let S = R[x] be the polynomial ring in one variable over R.
(a) Show that if a polynomial f is a zerodivisor in S, then there is an element r R cfw_0
such that rf = 0. [Sug
Math 614, Fall 2012
Due: Monday, December 17
Problem Set #6
This is an optional assignment. All problems are extra credit problems.
1. (a) Let S be a commutative ring and P1 , . . . , Pn be mutually incomparable prime ideals
n
of R. Let W = S i=1 Pi . Pro
Math 614, Fall 2012
Problem Set #6: Solutions
1.(a) Since W is the complement of a union of primes it is a multiplicative system, and
the primes of W 1 R correspond bijectively to the primes of R disjoint from W . This
n
also applies to 6. These are preci
Math 614, Fall 2012
Problem Set #5: Solutions
1. (a) If f g = 0 with g = 0, replace g by a multiple rg, r R, that is not 0 but has as
few nonzero terms as possible. Then every coecient of g has the same annihilator in R
as every other coecient: if one can
Math 614, Fall 2012
Due: Friday, November 16
Problem Set #4
1. Consider a system of m linear equations in n variables over a commutative ring R,
n
i=1 rij xi = rj , where 1 j m indexes the equations and the rij , rj are given elements
of R. Prove that the
Math 614, Fall 2012
Problem Set #4: Solutions
1. Let N be the submodule of Rm spanned by the n columns of the matrix, and let v
denote the column whose entries are the ri . The linear system has a solution i the v is
in the R-span of the columns of the ma