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1. Problem set 1: Modules, submodules, submodules generated by sets.

(1) Let R be a ring, M a left R-module, R = {Ni : i ∈ I} a family of submodules of M . Show that:

(a) T

i∈INi is a submodule of M , (b) S

i∈INi is a submodue of M if R is a chain.

IfS

i∈INi is a submodue of M , is R necessarily a chain?

(2) Let R be a ring, M a left R-module, let S ⊂ M . Show that:

hSi = {a1s1+ . . . + ansn+ b1t1 + . . . + bmtm : ai ∈ R, si ∈ S, bj ∈ Z, tj ∈ S}.

(3) Let R be a ring, M a left R-module, let N1, N2 < M . Show that hN1∪ N2i = {n1+ n2 : n1 ∈ N1, n2 ∈ N2}.

(4) Give an example of a finitely generated module, which is not a finitely generated Abelian group.

(5) Let R and S be rings, let φ : R → S be a ring homomorphism. Show that every S-module M is also an R-module by defining rm, r inR, m ∈ M as φ(r)m.

(6) Let R be a ring, I C R a two-sided ideal, M a left R-module. Show that M/IM is a R/I-module with (r + I)(a + IA) = ra + IA.

Homework: Problems 3,4,6 are to be handed in during the next class.

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