Math 8411, Homework #2
Due Friday September 21
1.(a) Prove that if (m, n) = 1 then ZmโZn= 0.
(b) Let AโBbe rings. Prove that if fโA[x] then the tensor product of
algebras A[x]/(f)โABis isomorphic to B[x]/(f).
(c) Use (b) to prove that CโRCis not a field. Is CโRCand integral domain?
2. (a) Prove that if AโBโCare rings such that Bis flat as an A-algebra,
and Cis flat as a B-algebra then Cis flat as an A-algebra.
(b) Prove that A[x1, . . . , xn] is flat as an Aalgebra. (Hint: Use induction to
reduce to the case of one variable.)
3. (a) Prove that if Bis flat as an A-module and IโAis an ideal then the
B-module IโABis isomorphic to the ideal IB. Can you give a counterexample if
Bis not flat?
(b) If Iis prime (resp. maximal) is IB necessarily prime (resp. maximal)?
(c) Prove that if Bis flat as an Aalgebra and IโA[x1, . . . , xn] is an ideal then
A[x1, . . . , xn]/I โABis isomorphic to B[x1, . . . , xn]/(IB [x1, . . . , xn]).
4. Let Rbe an integral domain and Mbe an R-module. Define the torsion
submodule T(M) = {mโM|rm = 0,for some r6= 0}. A module is torsion free if
T(M) = 0. This is a module version of the torsion subgroup of a finitely generated
abelian group.
(a) Prove that if Mis flat then Mis torsion free; i.e. T(M) = 0. (The converse
is false but it is harder to give examples.)
(b) Prove that if f:M0โMis an R-module homomorphism then f(T(M0)) โ
T(M) and that if 0 โM0โMโM00 is short exact sequence, then the sequence
0โT(M0)โT(M)โT(M00) is also short exact.
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