Exercises for Higher Algebra
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چکیده
(1) Let R be a division ring. Prove that every module over R is free. You will need to use Zorn’s lemma: Recall that a partially order set (=poset) S is a set with a relation x ≤ y defined between some pairs of elements x, y ∈ S, such that: (i) x ≤ x ; (ii) x ≤ y and y ≤ x implies x = y ; (iii) x ≤ y , y ≤ z ⇒ x ≤ z . A chain in S is a subset T ⊂ S such that for all t, t ′ in T , either t ≤ t ′ or t ′ ≤ t. We say that a chain has an upper bound if there’s an element s ∈ S (we don’t require s ∈ T ) such that s ≥ t for all t ∈ T . Zorn’s lemma states for a non-empty poset S that if every chain in S has an upper bound than S has a maximal element, namely an element s0 ∈ S such that if s ∈ S and s ≥ s0 then s = s0 (note that we do not require that s0 ≥ s for all s ∈ S). If you have never seen Zorn’s lemma in action, try to use it to prove that any ring R has a maximal left ideal. Take S to be the set of ideals I 6= R of R with the partial order I ≤ J if I ⊆ J.
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