let r be an associative ring with unity. an element a in r is said to be r-clean if a = e+r, where e is an idempotent and r is a regular (von neumann) element in r. if every element of r is r-clean, then r is called an r-clean ring. in this paper, we prove that the concepts of clean ring and r-clean ring are equivalent for abelian rings. further we prove that if 0 and 1 are the only idempotents...

let r be an associative ring with unity. an element a in r is said to be r-clean if a = e+r, where e is an idempotent and r is a regular (von neumann) element in r. if every element of r is r-clean, then r is called an r-clean ring. in this paper, we prove that the concepts of clean ring and r-clean ring are equivalent for abelian rings. further we prove that if 0 and 1 are the only idempotents...

In the title compound, C(20)H(16)S(2), the pyrene ring is planar [maximum deviation 0.0144 (15) Å] and the dithiane ring adopts a chair conformation. The crystal packing is stabilized by C-H⋯π inter-actions. An intra-molecular C-H⋯S hydrogen bond generates an S(5) ring motif.

In the title compound, C(16)H(18)N(2)O(2)S, the piperidine ring adopts a chair conformation. The central 4-thia-zolidinone ring makes dihedral angles of 12.01 (7) and 51.42 (9)°, respectively, with the benzene ring and the least-squares plane of the piperidine ring. An intra-molecular C-H⋯S hydrogen bond stabilizes the mol-ecular structure and generates an S(6) ring motif. In the crystal, mol-e...

In the title compound, C(22)H(24)N(4)S, the methylphenyl and isobutylphenyl rings are inclined at an angle of 79.98 (1)° and they form dihedral angles of 4.59 (1) and 75.47 (1)°, respectively, with the triazolothia-diazole unit. An intra-molecular C-H⋯S hydrogen bond generates an S(5) ring motif. The crystal structure is stabilized by inter-molecular C-H⋯N hydrogen bonds and weak C-H⋯π and π-π ...

in this work, we investigate the transfer of some homological properties from a ring $r$ to its amalgamated duplication along some ideal $i$ of $r$ $rbowtie i$, and then generate new and original families of rings with these properties.

In this paper we analyze and study the Smarandache idempotents (S-idempotents) in the ring Zn and in the group ring ZnG of a finite group G over the finite ring Zn. We have shown the existance of Smarandache idempotents (S-idempotents) in the ring Zn when n = 2 p (or 3p), where p is a prime > 2 (or p a prime > 3). Also we have shown the existance of Smarandache idempotents (S-idempotents) in th...

for a ring endomorphism $alpha$ and an $alpha$-derivation $delta$, we introduce a concept, so called skew $pi$-armendariz ring, that is a generalization of both $pi$-armendariz rings, and $(alpha,delta)$-compatible skew armendariz rings. we first observe the basic properties of skew $pi$-armendariz rings, and extend the class of skew $pi$-armendariz rings through various ring extensions. we nex...