Exercise 13, § 2

Let A be a principal ideal domain, let τ be an irreducible element of A, and let M be a π-primary module over A. Let I be the set of ordinals a (Set Theory, III, p. 225, Ex. 14) such that the cardinality of the set of ordinals < a is at most equal to that of M. Define submodules M^{(α)} of M for all a ∈ I by transfinite induction as follows: take M^{(0)} = M; take M^{(α + 1)} = πM^{(α)}; and if a is a limit ordinal take M^{(α)} to be the intersection of all the M^{(β)} for β < a.

a) Show that there exists a least ordinal τ ∈ I such that M^{(τ + 1)} = M^{(τ)} (the terminal ordinal of M). Show that M^{(τ)} admits a complement in M, and is the direct sum of submodules isomorphic to U_π (Ex. 3). Say that M is reduced if M^{(τ)} = 0. b) Let M be a reduced τ-module. For all x ∈ M, show that there exists a greatest ordinal γ(x) ≤ τ such that x ∈ M^{(γ(x))} (in particular, γ(0) = τ). Generalising the definition in Ex. 8, we say that γ(x) is the height of x in M. Show that if γ(x) < γ(y), then γ(x + y) = γ(x), and that if γ(x) = γ(y), then γ(x + y) ≥ γ(x). We say that x is proper with respect to a submodule N of M if for all y ∈ N the height γ(x + y) is at most equal to γ(x); show that then γ(x + y) = inf(γ(x), γ(y)), for all y ∈ N. c) Let a < τ be an ordinal, and let M^{(α)} be the submodule of M^{(α)} consisting of those x such that πx = πy for some y ∈ M^{(α + 1)}. For those y ∈ M^{(α + 1)} having this property, the elements of the form u = x − y form a class mod M(π) ∩ M^{(α − 1)} in M(τ) ∩ M^{(α)}; if φ is the canonical homomorphism from M(π) ∩ M^{(α)} onto the quotient of this module by M(π) ∩ M^{(α + 1)}, then show that the map ψ which sends x to φ(u) is a homomorphism with kernel M^{(α − 1)}, and consequently that M^{(α)}/M^{(α + 1)} is isomorphic to (M(π) ∩ M^{(α)})/(M(π) ∩ M^{(α + 1)}). d) Let N be a submodule of M, and let x ∈ M^{(α)} be an element of height α; show that x is proper with respect to N if and only if the corresponding elements u = x − y of M(π) ∩ M^{(α)} (cf. c)) are proper with respect to N. There exists such an element x if and only if the image of N ∩ M^{(α)} under the homomorphism ψ is distinct from (M(π) ∩ M^{(α)})/(M(π) ∩ M^{(α − 1)}).