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 Mp denote the elements of M killed by some power of p. Show that Mp 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.)
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 Mp denote the elements of M killed by some power of p. Show that Mp 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.)
posted by Allen Knutson @ 3:30 PM
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