the authors studied several conditions on an algebraic frame L. In particular, four properties called Reg(1), Reg(2), Reg(3), and Reg(4) were considered. There it was shown that Reg(3) is equivalent to the more familiar condition known as projectability. In this article we show that there is a nice property, which we call feebly projectable, that is between… (More)
Elementary divisor domains were defined by Kaplansky [I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. it was also proved that if a Hermite ring satisfies (N), then it is an elementary divisor ring. The aim of this article is to generalize this result (as well as others) to a much wider class of rings. Our main result is that Bézout rings… (More)
It is known that the free topological group over the Tychonoff space X, denoted F (X), is a P-space if and only if X is a P-space. This article is concerned with the question of whether one can characterize when F (X) is a weak P-space, that is, a space where all countable subsets are closed. Our main result is that F (X) is a weak P-space if and only if X… (More)
A ring is called clean if every element is the sum of a unit and an idempotent. Throughout the last 30 years several characterizations of commutative clean rings have been given. We have compiled a thorough list, including some new equivalences, in hopes that in the future there will be a better understanding of this interesting class of rings. One of the… (More)
An element in a ring is called clean if it may be written as a sum of a unit and idempotent. The ring itself is called clean if every element is clean. Recently, Anderson and Camillo (Anderson, D. D., Camillo, V. (2002). Commutative rings whose elements are a sum of a unit and an idempotent. Comm. Algebra 30(7):3327–3336) has shown that for commutative… (More)
When every projective module is a direct sum of finitely generated modules ✩ Abstract We characterize rings over which every projective module is a direct sum of finitely generated modules, and give various examples of rings with and without this property.
A commutative ring A is said to be clean if every element of A can be written as a sum of a unit and an idempotent. This definition dates back to 1977 where it was introduced by W. investigated commutative clean rings and obtained several important results. In  Han and Nicholson show that if A is a semiperfect ring, then A[Z 2 ] is a clean ring. In this… (More)
In this article the frame-theoretic account of what is archimedean for order-algebraists, and semisimple for people who study commutative rings, deepens with the introduction of J-frames: compact normal frames that are join-generated by their saturated elements. Yosida frames are examples of these. In the category of J-frames with suitable skeletal… (More)
The functor on the category of bounded lattices induced by reversing their order, gives rise to a natural equivalence of coherent frames. We investigate the spectra as well as some well-known frame properties like zero-dimensionality and normality.
Bazzoni's Conjecture states that the Prüfer domain R has finite character if and only if R has the property that an ideal of R is finitely generated if and only if it is locally principal. In  the authors use the language and results from the theory of lattice-ordered groups to show that the conjecture is true. In this article we supply a purely ring… (More)