12 research outputs found
Noncommutative generalizations of theorems of Cohen and Kaplansky
This paper investigates situations where a property of a ring can be tested
on a set of "prime right ideals." Generalizing theorems of Cohen and Kaplansky,
we show that every right ideal of a ring is finitely generated (resp.
principal) iff every "prime right ideal" is finitely generated (resp.
principal), where the phrase "prime right ideal" can be interpreted in one of
many different ways. We also use our methods to show that other properties can
be tested on special sets of right ideals, such as the right artinian property
and various homological properties. Applying these methods, we prove the
following noncommutative generalization of a result of Kaplansky: a (left and
right) noetherian ring is a principal right ideal ring iff all of its maximal
right ideals are principal. A counterexample shows that the left noetherian
hypothesis cannot be dropped. Finally, we compare our results to earlier
generalizations of Cohen's and Kaplansky's theorems in the literature.Comment: 41 pages. To appear in Algebras and Representation Theory. Minor
changes were made to the numbering system, in order to remain consistent with
the published versio
When is the Intersection of Two Finitely Generated Subalgebras of a Polynomial Ring Also Finitely Generated?
Per2-Mediated Vascular Dysfunction Is Caused by the Upregulation of the Connective Tissue Growth Factor (CTGF)
LR-90 prevents methylglyoxal-induced oxidative stress and apoptosis in human endothelial cells
Microvascular lesions of diabetic retinopathy: clues towards understanding pathogenesis?
Zero-divisor graphs in commutative rings
This article surveys the recent and active area of zero-divisor graphs of commutative rings. Notable algebraic and graphical results are given, followed by a historical overview and an extensive bibliography