A ring R is called strongly clean if every element of R is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey [3] completely characterized the commutative local rings R for which Mn(R) is strongly clean. For a general local ring R and n > 1, however, it is unknown when the matrix ring Mn(R) is strongly clean. Here we completely determi...

We show that if $R$ is a ring with an arbitrary idempotent $e$ such that $eRe$ and $(1-e)R(1-e)$ are both strongly nil-clean rings‎, ‎then $R/J(R)$ is nil-clean‎. ‎In particular‎, ‎under certain additional circumstances‎, ‎$R$ is also nil-clean‎. ‎These results somewhat improves on achievements due to Diesl in J‎. ‎Algebra (2013) and to Koc{s}an-Wang-Zhou in J‎. ‎Pure Appl‎. ‎Algebra (2016)‎. ‎...

A series of ruthenium olefin metathesis catalysts of the general structure (H(2)IMes)(PR(3))(Cl)(2)Ru=CHPh (H(2)IMes = 1,3-dimesityl-4,5-dihydroimidazol-2-ylidene) have been prepared; these complexes are readily accessible in two steps from commercially available (H(2)IMes)(PCy(3))(Cl)(2)Ru=CHPh. Their phosphine dissociation rate constants (k(1)), relative rates of phosphine reassociation, and ...

in this paper, we introduce the new notion of n-f-clean rings as a generalization of f-clean rings. next, we investigate some properties of such rings. we prove that mn(r) is n-f-clean for any n-f-clean ring r. we also, get a condition under which the denitions of n-cleanness and n-f-cleanness are equivalent.

We report on the block copolymerization of two structurally different norbornene monomers (±)-endo,exo-bicyclo[2.2.1]-hept-5-ene-2,3-dicarboxylic acid dimethylester (7), and (±)-endo,exo-bicyclo[2.2.1]-hept-5-ene-2,3-dicarboxylic acid bis(1-oxyl-2,2,6,6-tetramethyl-piperidin-4-yl) ester (9) using ruthenium based Grubbs' type initiators [(PCy₃)₂Cl₂Ru(benzylidene)] G1 (PCy₃ = tricyclohexylphosphi...

A ring R is called strongly clean if every element of R is the sum of a unit and an idempotent that commute. By SRC factorization, Borooah, Diesl, and Dorsey [3] completely determined when Mn(R) over a commutative local ring R is strongly clean. We generalize the notion of SRC factorization to commutative rings, prove that commutative n-SRC rings (n ≥ 2) are precisely the commutative local ring...

A square matrix is said to be alternating-clean if it is the sum of an alternating matrix and an invertible matrix. In this paper, we determine all alternating-clean matrices over any division ring K. If K is not commutative, all matrices are alternating-clean, with the exception of the 1× 1 zero matrix. If K is commutative, all matrices are alternating-clean, with the exception of odd-size alt...

A series of bis-carbene Ru-hydride species, including (IMes)(Im(OMe)2)(PPh3)RuHCl (1) and (SIMes)(Me2Im(OMe)2)(PPh3)RuHCl (2) were prepared and subsequently shown to react with aryl-vinyl-sulfides to give the bis-carbene-alkylidene complexes: Im(OMe)2(SIMes)RuCl(SR)(=CHCH3) (R = p-FC6H4 (3), p-(NO2)C6H4 (4)), Im(OMe)2(IMes)RuCl(=CHCH3)(SPh) (5), Me2Im(OMe)2(SIMes)RuCl(=CHCH3)(SPh) (6), Im(OMe)2...