Annihilator-stability and unique generation in C(X)

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].

Strongly unit nil-clean rings

An element in a ring is strongly nil-clean, if it is the sum of an idempotent and a nilpotent element that commute. A ring [Formula: see text] is strongly unit nil-clean, if for any [Formula: see text] there exists a unit [Formula: see text], such that [Formula: see text] is strongly nil-clean. We prove, in this paper, that a ring [Formula: see text] is strongly unit nil-clean, if and only if every element in [Formula: see text] is equivalent to a strongly nil-clean element, if and only if for any [Formula: see text], there exists a unit [Formula: see text], such that [Formula: see text] is strongly [Formula: see text]-regular. Strongly unit nil-clean matrix rings are investigated as well.

A nil-clean 2 × 2 matrix over the integers which is not clean

While any nil-clean ring is clean, the last eight years, it was not known whether nil-clean elements in a ring are clean. We give an example of nil-clean element in the matrix ring ℳ2(Z) which is not clean.