Math 614, Fall 2008
Supplementary Problem Set #2: Solutions
1. (a) We have a composite map R S W 1 S and the image of V is in the units
of W 1 S. This gives the existence of a unique map as required. Alternatively, since
(r/v)(v/1) = r/1, we nd we must ha
Math 614, Fall 2008
Supplementary Problem Set #3: Solutions
1. Let M denote the ideal generated by all the monomials of R = R of positive degree.
Well show that M is not nitely generated. Suppose (mi , ni ), 1 i h are such that the
i = xmi y ni are genera
Math 614, Fall 2008
Problem Set #3: Solutions
1. We have that u is represented as c/f m and v as d/g n . Since these become equal in
Rf g , we can choose k such that (f g)k (g n c f m d) = 0. Since f, g generate the unit ideal,
so do F = f m+k , G = g n+k
Math 614, Fall 2008
Problem Set #2: Solutions
1. In the category of sets, the empty set is an initial object and a set with one element
is a nal object. In the category of commutative rings with identity,Z is an initial object
(the unique map Z R sends n
Math 614, Fall 2008
Due: Friday, September 26
Problem Set #1
In #1. and #2. K is a eld and R = K[x] is a polynomial ring in one variable over K.
1. (a). Show that if h, k > 0 are integers with greatest common divisor d, then the subring
K[xh , xk ] of R g
Math 614, Fall 2008
Problem Set #1: Solutions
1. (a) The ring generated is spanned over K by cfw_xt : t T where T is the set of all
integers of the form mh + nk for m, n N. Evidently, this set is contained in dN, the
nonnegative multiples of d. It suces
Math 614, Fall 2008
Problem Set #4: Solutions
1. S has the K-basis B = cfw_y n : n N cfw_xh y k : h 1, k Z. B0 = B cfw_1 is a K-basis
for m S, and xB0 = cfw_xy n : n 1 cfw_xh y k : h 2, k Z is a K-basis for J. Hence,
the image B of cfw_y n : n N cfw_xy k
Math 614, Fall 2008
Problem Set #5: Solutions
1. Let m1 , . . . , mh and n1 , . . . , nk generate M and N respectively. Then the mi nj
generate M R N , and the map HomR (M, N ) N h that sends f f (m1 ), . . . , f (mh )
is injective, so that HomR (M, N ) N
Math 614, Fall 2008
Supplementary Problem Set #5: Solutions
1. Compare each of two sets of generators with their union. It thus suces to do the case
where one, u1 , . . . , un , is contained in the other, u1 , . . . , un+k . By induction on k, we may
n
as
Math 614, Fall 2008
Supplementary Problem Set #1: Solutions
1. (a) This is clear if R = K. If not, choose f R K of degree d > 0, and replace f
by a scalar multiple so that it has leading coecient 1: say f = xd + cd1 xd1 + + c0 .
Then xd + cd1 xd1 + + c0 f
Math 614, Fall 2008
Supplementary Problem Set #4: Solutions
1. If u frac(T ) then u frac(Si ) for all i. If u is integral over T , it is integral over every
Si , and therefore in every Si . But then u T .
2. This ring is not Noetherian. Consider the decre