Annihilator-stability and unique generation in C(X).

Using the equivalence of unique generation and cleanness of C (X) , 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 C (X) is UG (uniquely generated) then M n (C (X)) is too, and R is strongly associate if and only if R [ [ x ] ] is, where R = C (X). We give topological characterizations of UG and AS (annihilator-stable) elements of C (X) for continuum spaces X and using this, we observe that the product of two UG elements need not be UG. It is shown that the set of elements in C (X) which have stable range 1 and the set of AS elements of C (X) coincide and several examples are given which show that the set of UG elements, the set of AS elements and the set of clean elements of C (X) can differ. Finally, we characterize spaces X for which every clean element of C (X) or every element of C (X) which has stable range 1 is UG. [ABSTRACT FROM AUTHOR]