substitution of two or four carbon atoms by nitrogen in the corannulene molecule as a carbon nanostructure was done and the obtained structures were optimized at mp2/6-31g(d) level of theory. calculations of the nucleus-independent chemical shift (nics) were performed to analyze the aromaticity of the corannulene rings and its derivatives upon doping with n at b3lyp/6-31g(d) level of theory. re...

We investigate the sufficient conditions and the necessary conditions on an exchange ring R under which R has stable range one. These give nontrivial generalizations of Theorem 3 of V. P. Camillo and H.-P. Yu (1995), Theorem 4.19 of K. R. Goodearl (1979, 1991), Theorem 2 of R. E. Hartwig (1982), and Theorem 9 of H.-P. Yu (1995). 2000 Mathematics Subject Classification. Primary 16E50, 19B10. An ...

‎‎let r be a non-commutative ring with unity‎. ‎the commuting graph‎ ‎of $r$ denoted by $gamma(r)$‎, ‎is a graph with a vertex set‎ ‎$rsetminus z(r)$ and two vertices $a$ and $b$ are adjacent if and only if‎ ‎$ab=ba$‎. ‎in this paper‎, ‎we investigate non-commutative rings with unity of order $p^n$ where $p$ is prime and $n in lbrace 4,5 rbrace$‎. it is shown that‎, ‎$gamma(r)$ is the disjoint ...

Let R be a commutative ring. An R-module M is called co-multiplication provided that foreach submodule N of M there exists an ideal I of R such that N = (0 : I). In this paper weshow that co-multiplication modules are a generalization of strongly duo modules. Uniserialmodules of finite length and hence valuation Artinian rings are some distinguished classes ofco-multiplication rings. In additio...

Throughout this paper R denotes an associative ring with identity and all modules are unitary. We use the symbol U(R) to denote the group of units of R and Id(R) the set of idempotents of R, Un(R) the set of elements which are the sum of n units of R, UΣ(R) the set of elements each of which is the sum of finitely many units in R, RE(R) (URE(R)) the set of regular (unit regular) elements of R, a...