On a Question of Hartwig and Luh
نویسندگان
چکیده
In 1977 Hartwig and Luh asked whether an element a in a Dedekind-finite ring R satisfying aR = a2R also satisfies Ra = Ra2. In this paper, we answer this question in the negative. We also prove that if a is an element of a Dedekind-finite exchange ring R and aR = a2R, then Ra = Ra2. This gives an easier proof of Dischinger’s theorem that left strongly π-regular rings are right strongly π-regular, when it is already known that R is an exchange ring. 2010 Mathematics subject classification: primary 16E50; secondary 16U99.
منابع مشابه
Efficient buchwald hartwig reaction catalyzed by spions-bis(NHC)-Pd(II)
A powerful and convenient reaction procedure for the C-N coupling reaction (the Buchwald-Hartwig reaction), yielding products of N-arylanilines and N-arylamines in both conventional heating and microwave irradiation has been reported. The protocol utilizes a stable and new supper ferromagnetic nanoparticle chelating N-heterocyclic dicarbene palladium(II) complex (Pd-NHC) as catalyst which helps...
متن کاملSPION-A-Pd(EDTA) (superparamagnetic iron oxide nanoparticles-A-ethylendiamine tetraacetic acid) as a robust nanocatalyst for Buchwald-Hartwig C-N Coupling Reaction
During thelast20-30 years, palladium-catalyzed reactions have witnessed tremendous advances in the industrial and organic reactions such as hydrogenation, coupling, cyanation and amination. Despite the wide utility of Pd-catalysts in these reactions, they suffer from a number of drawbacks such as recovery, reuse of catalyst and remain as a contaminant in the products at the end of the reaction...
متن کاملSynthesis of Symmetrical Triaryl Amines by Nano-CuO Catalyzed Buchwald-Hartwig Cross-coupling Reaction: NH2-Thiadiazole as a New N-source
The one-pot synthesis of symmetrical triaryl amines using aryl halides and 5-Methyl-1,3,4-Thiadiazole-2-Amine as a nitrogen-transfer reagent is reported. Copper oxide nanoparticles that does not require the presence of any additional ligand, improved rate and facility of the synthesis of triaryl amines. The introduction of a new N-source, using green solvents PEG/H2O, normal atmospheric conditi...
متن کاملFunctional Analysis of Three Arabidopsis Transcriptional Co- Repressors Linking Floral Regulators to Aba Hormone Signaling
Title of Dissertation: FUNCTIONAL ANALYSIS OF THREE ARABIDOPSIS TRANSCRIPTIONAL COREPRESSORS LINKING FLORAL REGULATORS TO ABA HORMONE SIGNALING Jayashree Sitaraman, Doctor of Philosophy, 2004 Dissertation Directed By: Associate Professor Zhongchi Liu Department Of Cell Biology and Molecular Genetics Hormonal signaling and developmental patterning have mostly been studied as separate and indepen...
متن کاملThe transcriptional regulator LEUNIG_HOMOLOG regulates mucilage release from the Arabidopsis testa.
Exposure of the mature Arabidopsis (Arabidopsis thaliana) seed to water results in the rapid release of pectinaceous mucilage from the outer cells of the testa. Once released, mucilage completely envelops the seed in a gel-like capsule. The physical force required to rupture the outer cell wall of the testa comes from the swelling of the mucilage as it expands rapidly following hydration. In th...
متن کامل