Topics in algebra Chapter IV: Commutative rings and modules I - 1

4. Hom(Zn, Zm) ∼ = Zd, where d is g.c.d. of m and n.
5. The sequence 0 −→ N1
φ
−→ N2
R-module M, the sequence 0 −→ HomR(M, N1)
HomR(M, N3) is exact.
6. The sequence N1
φ
−→ N2
R-module M, the sequence 0 −→ HomR(N3, M)
HomR(N1, M) is exact.
ψ
ψ
−→ N3 is exact if and only if for every
˜φ
−→ HomR(M, N2)
˜ψ
−→
−→ N3 −→ 0 is exact if and only if for every
˜ψ
−→ HomR(N2, M)
7. Let M and N be R-modules; then M ⊗R N ∼ = N ⊗R M.
8. Prove that free modules are flat.
˜φ
−→
9. Let M be an R-module and I be an ideal of R; then M ⊗R R/I ∼ =
M/IM.
10. Let I and J be ideals of R; then R/I ⊗R R/J ∼ = R/(I + J).
11. Zm ⊗ Zn ∼ = Zd, where d is g.c.d. of m and n.
12. Let G be a torsion abelian group; then G ⊗ Q = 0.
13. Let M be an R-module and Mm = 0 for every maximal ideal m of R;
then M = 0.
14. Let f : N −→ M be an R-homomorphism and fm : Nm −→ Mm is an
injection for every maximal ideal m of R; then f is an injection.
15. Q ⊗ Q ∼ = Q.
16. If P is a projective module, then PS is a projective RS module, where
S is a multiplicative subset of R.
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