Exercise B5

1. Prove that every quotient of a divisible R-module is again a divisible R-module. Let M be a divisible R-module and let N ⊆ M be a submodule of M. If we take an element \overline{x} ∈ M/N, then x ∈ M. By hypothesis, for all r ∈ R, r ≠ 0, there exists y ∈ M such that x = ry. Since \overline{x} = \overline{ry} = r\overline{y}, it follows that M/N is again a divisible R-module.

2. If R is a PID, then every quotient of an injective R-module is again an injective R-module. We can use the following Theorem (pg. 22 of Tony’s Notes): if R is a PID, then the R-module M is injective if and only if it is a divisible R-module. So

M injective ⟺ M divisible ⟹^{1.} M/N divisible ⟺ M/N injective