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Let R be a ring

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Algebra I. Homework 5. Due on October 15, 2020.

1. Let R be a ring.

(a) Give an example of two submodules of an R-module whose union is not a submodule;

(b) If {Mn} is an increasing family of submodules of an R-module M, i.e.

Mn ⊂Mp forn ≤p, show that ∪Mn is a submodule of M.

2. Let R be an integral ring, let K be its field of fractions. Assume K 6= A.

Prove or disprove: K is a free A-module.

3. Classify all modules of finite type over Z/4Z.

4. Let α be a complex root of the irreducible polynomial x3−3x+ 4. Find the inverse of α2+α+ 1 in Q(α) explicitly, in the forma+bα+cα2, a, b, c∈Q.

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