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Positive roots suffice for divisibility
PDF p. 13: Divisibility of the Picard group
Statement
An abelian group is divisible if and only if, for every positive integer n, multiplication by n is surjective.
An abelian group is written additively. Multiplication by n means repeated addition, extended to negative integers using additive inverses. A map is surjective if every element of its codomain has a preimage.
Facts & Assumptions
A divisible abelian group is a divisible integer module: for every nonzero integer m and every element a, there is an element b with mb = a (Divisible modules over an integral domain).
Given: An abelian group A.
Goal: Prove the stated equivalence.
Proof
If A is divisible, apply [F1] with for each positive to obtain surjectivity.
Conversely, assume each positive multiplication map is surjective. For nonzero m and a in A, choose b with . If m is positive use b; if m is negative use -b. In either case the chosen element maps to a under multiplication by m. Thus [F1] gives divisibility.
Depends on
Dependency tree · 3 levels
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