In this paper,we investigate the comparability structure over exchange rings.It is shown that the subdirect product of an exchange ring with stable range one and an exchange ring satisfying the comparability is also a...In this paper,we investigate the comparability structure over exchange rings.It is shown that the subdirect product of an exchange ring with stable range one and an exchange ring satisfying the comparability is also an exchange ring satisfying the comparability.This provides a new class of exchange rings satisfying the comparability.Furthermore,we investigate the s-comparability over exchange rings.This generalizes the corresponding results of Goodearl and Chen.展开更多
We obtaim a new substitution for modules over exchange rings satisfving related compara- bilitv. Also we investigate the structure of modules over exchange rings satisfying power comparability and provide a new class ...We obtaim a new substitution for modules over exchange rings satisfving related compara- bilitv. Also we investigate the structure of modules over exchange rings satisfying power comparability and provide a new class of exchange rings satisfying related comparability.展开更多
The localized molecular orbitals and energy levels for four typical conjugated six-membered ring systems C_6H_6, C_3N_3H_3, B_3N_3H_6, and (B_3O_6)^(3-) as well as a non-aromatic reference molecule N_3Cl_6 have been ...The localized molecular orbitals and energy levels for four typical conjugated six-membered ring systems C_6H_6, C_3N_3H_3, B_3N_3H_6, and (B_3O_6)^(3-) as well as a non-aromatic reference molecule N_3Cl_6 have been calculated by using Edmiston-Ruedenberg energy localization technique under the CNDO/2 approximation in order to investigate the nature of aromaticity or quasi-aromaticity of the six-membered ring systems studied. The contour maps for π-type localized MO's (LMO) have been plotted to illustrate the bonding characteristics of the five ring systems studied. These LMO calculations show that for all the conjugated six-membered ring systems considered there exists local delocalization of π-bonds or three-centered and occasionally four-centered two-electron π-bonds in our terminology, and the cooperative effect among these π-bonds leading to the formation of a closed continuous x-conjugation system around the ring, which is necessary for the creation of aromaticity in the systems studied. We have been able to discuss the properties of these three-centered π-bonds in terms of the constituent atoms and electrons and the relevant orbitals involved.展开更多
基金This work is supported by the National Natural Science Foundation of China (Grant No.19801012)the Ministry of Education of China.
文摘In this paper,we investigate the comparability structure over exchange rings.It is shown that the subdirect product of an exchange ring with stable range one and an exchange ring satisfying the comparability is also an exchange ring satisfying the comparability.This provides a new class of exchange rings satisfying the comparability.Furthermore,we investigate the s-comparability over exchange rings.This generalizes the corresponding results of Goodearl and Chen.
基金This work is supported by the National Natural Science Foundation of China (Grant No. 19801012. 19531020).
文摘We obtaim a new substitution for modules over exchange rings satisfving related compara- bilitv. Also we investigate the structure of modules over exchange rings satisfying power comparability and provide a new class of exchange rings satisfying related comparability.
文摘The localized molecular orbitals and energy levels for four typical conjugated six-membered ring systems C_6H_6, C_3N_3H_3, B_3N_3H_6, and (B_3O_6)^(3-) as well as a non-aromatic reference molecule N_3Cl_6 have been calculated by using Edmiston-Ruedenberg energy localization technique under the CNDO/2 approximation in order to investigate the nature of aromaticity or quasi-aromaticity of the six-membered ring systems studied. The contour maps for π-type localized MO's (LMO) have been plotted to illustrate the bonding characteristics of the five ring systems studied. These LMO calculations show that for all the conjugated six-membered ring systems considered there exists local delocalization of π-bonds or three-centered and occasionally four-centered two-electron π-bonds in our terminology, and the cooperative effect among these π-bonds leading to the formation of a closed continuous x-conjugation system around the ring, which is necessary for the creation of aromaticity in the systems studied. We have been able to discuss the properties of these three-centered π-bonds in terms of the constituent atoms and electrons and the relevant orbitals involved.
