Knot modules and the elementary divisor theorem

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We show that a knot module M is a cyclic module if and only if E 1( M)= Λ, all Steinitz-Fox-Smythe invariants of quotients of M by irreducible factors of Δ 0( M) are trivial, and certain unit class invariants describing the extensions in a composition series for M are trivial. (We give several counter-examples to illustrate the independence of these conditions.) We show also that a direct sum of cyclic modules (over any noetherian factorial domain) satisfies the Elementary Divisor Theorem if and only if all its elementary ideals are principal, and we replace the Dedekind condition in Levine's π-primary sequence realization theorem by a (weaker) projectivity condition.