Annihilator-stability and unique generation in C(X)

Abstract

Using the equivalence of unique generation and cleanness of [Formula: see text], we give affirmative answers to questions raised in [I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc.  66 (1949) 464–491] and [D.D. Anderson et al., When are associates unit multiples? Rocky Mountain J. Math.  34 (2004) 811–828] for rings of real-valued continuous functions. In fact, we show that if [Formula: see text] is [Formula: see text] (uniquely generated) then [Formula: see text] is too, and [Formula: see text] is strongly associate if and only if [Formula: see text] is, where [Formula: see text]. We give topological characterizations of [Formula: see text] and [Formula: see text] (annihilator-stable) elements of [Formula: see text] for continuum spaces [Formula: see text] and using this, we observe that the product of two UG elements need not be [Formula: see text]. It is shown that the set of elements in [Formula: see text] which have stable range 1 and the set of [Formula: see text] elements of [Formula: see text] coincide and several examples are given which show that the set of [Formula: see text] elements, the set of [Formula: see text] elements and the set of clean elements of [Formula: see text] can differ. Finally, we characterize spaces [Formula: see text] for which every clean element of [Formula: see text] or every element of [Formula: see text] which has stable range 1 is [Formula: see text].