Final exam

The final exam is take-home, open [Hungerford] and notes, and THREE HOURS. After you first look at it, you're done three hours later. If you're in the class, you can't discuss it with people who have taken or are taking it, until you yourself have finished taking it.

You can download it here but probably best not until you're ready to take it.

When you're done, turn it in to the front office before they close on Friday. (Don't tempt fate, eh?) Then check "has everyone within earshot and 200B taken the final?" before discussing it.

Obviously there are tricky ways to circumvent the spirit of this law while adhering to the letter, and I count on you to be clever enough to think of them. Don't do those either. That ought to cover it I think.

HW due Fri Mar 9 (2c corrected)

1. Let R->S be a ring homomorphism such that S is a finitely generated R-module. Let M be a finitely generated S-module. (Example: the complex numbers over the reals. Non-example: R[x] as a module over R.)
Show that M is a finitely generated R-module. (Example: finite-dim complex vector spaces are finite real-dim too.)

2. Let R = C[x,y] / < xy > (not < x,y > as I miswrote at first). Let Rⁿ be the usual free module.

a) Find a non-principal ideal of R.

b) Nonetheless, show that any submodule of Rⁿ is finitely generated.

c) Let A be an n x k matrix with entries from R, where k is very big ( > 2n, maybe even infinite).
i. Show that by invertible row and column operations, we can reduce A to having at most 2n nonzero columns.
ii. Further, v1: one can in addition ask the left n columns are [a diagonal matrix involving x and no y] + [a matrix involving y and no x], and the next n columns involve y and no x.
iii. Further, v2: one can in addition to (i) ask that the second n columns are [a diagonal matrix involving y and no x].

HINT: Prove 2c(ii) or 2c(iii), which proves 2c(i) and from there 2b.

3. Say R is a subring of a domain S,
a,b in S,
p(x) = x³ + p₂x² + p₁x + p₀,
q(x) = x³ + q₂x² + q₁x + q₀,
p(a),q(b)=0.
Construct R-polynomials s,t with leading coefficient 1 such that s(a+b)=0, t(ab)=0.

N.B. I will grudgingly accept an abstract construction rather than the actual polynomials. Maybe 3 and 3 were already too big. : -8 (

4a. Let R be a ring, I an ideal, and P a prime ideal containing I.
Show that P contains the radical of I, denoted rad(I) for the rest of this question.

b. Assume R is a PID, and P = rad(I) is prime. Show that I is a power of P.

c. Assume R = C[x,y]/< xy >, and I = < x + cy > for c nonzero.

i. Show that rad(I) = < x,y >, and that this ideal is prime. Call it P.

ii. Compute the C-dimensions of R/I, R/P, R/P². Show that I is not a power of P.

5a. Let R be a PID. Show that the set of ideals with a given radical is countable.

b. Show that the ideals < x + cy > from 4c are all different, i.e. uncountably many ideals with the same radical.

6. Let R be a PID contained in S, and S finitely generated as an R-module.
Let M be a finitely generated S-module.

a. If M is a torsion S-module, does that imply that M is a torsion R-module?

b. Show that multiplication by any s in S gives a map M->M that is a R-module homomorphism.

c. Let p be a prime element of R, and let M_p denote the elements of M killed by some power of p. Show that M_p is not only an R-submodule of M, but an S-submodule. (This is most interesting when M is a torsion R-module; see if you can guess why.)