## M953 Homework 2 (Click
here
for solutions)

Due Friday, February 8, 2002

[1] Let S be the ring **C**[x,y] of complex
polynomials in two variables, and let R be the ring
**R**[x,y] of real
polynomials in two variables. Let f = y + xy - x^{2}
and g = y^{2} + xy^{2} - x^{3}, let J be
the ideal (f,g) in S, and let I be
the ideal (f,g) in R.
- (a)
- (i) Determine V(J) in
**A**(**C**^{2}).
- (ii) Conclude that Rad(J) = (x,y), where (x,y) means the ideal
in S generated by x and y.

- (b)
- (i) Explicitly find an n and m such that
x
^{m} and y^{n} are in I.
- (ii) Conclude that Rad(I) = (x,y), where now (x,y) means the ideal
in R generated by x and y.

[2] If k is an algebraically closed field, use the Weak Nullstellensatz
to show that every maximal ideal of k[x_{1}, ... ,x_{n}]
is of the form (x_{1} - a_{1}, ... ,
x_{n} - a_{n}), for some elements a_{i} of k.

[3] The Weak Nullstellensatz says that when k is
an algebraically closed field, the following statement holds:
"If J is a proper ideal of
k[x_{1}, ... ,x_{n}], then V(J) is nonempty."
When n = 1, show that the quoted statement is equivalent to
saying "Every nonconstant polynomial in k[x_{1}]
has a root." [You may assume that k[x_{1}] is a PID.]

[4] Let R be a commutative ring and let I and J be ideals in
the polynomial ring R[x]
such that I contains J. For each
element f of I, assume that there is an element g of J such that
f and g have the same leading coefficient, and deg(f) >= deg(g).
Show that then J must equal I.

[5] (This one is for 901-902 people, but everyone of course can try it.)
Hilbert was interested in finite generation for rings. The case he was
interested in involved a subring R = k[g_{1}, g_{2}, ...]
of S = k[x_{1}, ... ,x_{n}] generated by forms
g_{1}, g_{2}, ..., etc., in S of positive degree.
This problem shows why this is related to R being Noetherian
(i.e., its ideals are finitely generated). So
let I = (g_{1}, g_{2}, ...) be the ideal in R
generated by the forms. Show that the following are equivalent:
- (a) R is finitely generated.
- (b) R is a Noetherian ring.
- (c) I is finitely generated.