CS Ring Research Articles

C-Injective rings

A ring [Formula: see text] is called a right C-injective ring if every homomorphism from a closed right ideal of [Formula: see text] to [Formula: see text] can be extended to one from [Formula: see text] to [Formula: see text]. It is clear that a right CS ring must be right C-injective. Left C-injective rings can be defined similarly. Properties of C-injective rings are explored in this paper. It is shown that a left C-injective ring may not be right C-injective and a right C-injective ring may not be right CS. Some extensions of C-injective rings are discussed.

Weakly-$$\tau $$-Flat Modules

In this paper, we introduce and study a torsion-theoretic generalization of the weakly-flat module using the concept of $$\tau $$ -closed submodule for an idempotent radical $$\tau $$ . We introduce a new class of modules called weakly- $$\tau $$ -flat. We study these modules, generalizing several results both on extending modules and flat modules. We study generalizations and characterizations of IF ring, (finitely) $$\varSigma $$ -CS ring, and C-ring by weakly- $$\tau $$ -flat modules.

Neat-flat Modules

Let R be a ring. A right R-module M is said to be neat-flat if the kernel of any epimorphism Y → M is neat in Y, i.e., the induced map Hom(S, Y) → Hom(S, M) is surjective for any simple right R-module S. Neat-flat right R-modules are projective if and only if R is a right -CS ring. Every cyclic neat-flat right R-module is projective if and only if R is right CS and right C-ring. It is shown that, over a commutative Noetherian ring R, (1) every neat-flat module is flat if and only if every absolutely coneat module is injective if and only if R ≅ A × B, wherein A is a QF-ring and B is hereditary, and (2) every neat-flat module is absolutely coneat if and only if every absolutely coneat module is neat-flat if and only if R ≅ A × B, wherein A is a QF-ring and B is Artinian with J 2(B) = 0.

A Note on the Faith–Menal Conjecture

A ring R is called right Johns if R is right noetherian and every right ideal of R is a right annihilator. R is called strongly right Johns if the matrix ring M n (R) is right Johns for each integer n ≥ 1. The Faith–Menal conjecture is an open conjecture on QF rings. It says that every strongly right Johns ring is QF. It is proved that the conjecture is true if every closed left ideal of the ring R is finitely generated. This result improves the known result that the conjecture is true if R is a left CS ring.

Absolutely s-Pure Modules and Neat-Flat Modules

Let R be a ring with an identity element. We prove that R is right Kasch if and only if injective hull of every simple right R-modules is neat-flat if and only if every absolutely pure right R-module is neat-flat. A commutative ring R is hereditary and noetherian if and only if every absolutely s-pure R-module is injective and R is nonsingular. If every simple right R-module is finitely presented, then (1) R R is absolutely s-pure if and only if R is right Kasch and (2) R is a right -CS ring if and only if every pure injective neat-flat right R-module is projective if and only if every absolutely s-pure left R-module is injective and R is right perfect. We also study enveloping and covering properties of absolutely s-pure and neat-flat modules. The rings over which every simple module has an injective cover are characterized.

Classes of Almost Clean Rings

A ring is clean (almost clean) if each of its elements is the sum of a unit (regular element) and an idempotent. A module is clean (almost clean) if its endomorphism ring is clean (almost clean). We show that every quasi-continuous and nonsingular module is almost clean and that every right CS (i.e. right extending) and right nonsingular ring is almost clean. As a corollary, all right strongly semihereditary rings, including finite AW *-algebras and noetherian Leavitt path algebras in particular, are almost clean. We say that a ring R is special clean (special almost clean) if each element a can be decomposed as the sum of a unit (regular element) u and an idempotent e with aR ∩ eR = 0. The Camillo-Khurana Theorem characterizes unit-regular rings as special clean rings. We prove an analogous theorem for abelian Rickart rings: an abelian ring is Rickart if and only if it is special almost clean. As a corollary, we show that a right quasi-continuous and right nonsingular ring is left and right Rickart. If a special (almost) clean decomposition is unique, we say that the ring is uniquely special (almost) clean. We show that (1) an abelian ring is unit-regular (equiv. special clean) if and only if it is uniquely special clean, and that (2) an abelian and right quasi-continuous ring is Rickart (equiv. special almost clean) if and only if it is uniquely special almost clean. Finally, we adapt some of our results to rings with involution: a *-ring is *-clean (almost *-clean) if each of its elements is the sum of a unit (regular element) and a projection (self-adjoint idempotent). A special (almost) *-clean ring is similarly defined by replacing “idempotent” with “projection” in the appropriate definition. We show that an abelian *-ring is a Rickart *-ring if and only if it is special almost *-clean, and that an abelian *-ring is *-regular if and only if it is special *-clean.

Palladium(II) and Platinum(II) Complexes of N-Phenyl- and N-Ethyl-N′-pyrimidin-2-ylthiourea

Palladium(II) and platinum(II) complexes of N-ethyl-N′-pyrimidin-2-ylthiourea(HL1) and N-phenyl-N′-pyrimidin-2-ylthiourea (HL2) have been prepared, and the complexes [M(HL)Cl2], [Pt(L)2], [Pd(HL1)2]Cl2, and [Pd(L2)2] (where M = PdII or PtII) were characterized. The spectroscopic data are consistent with coordination of thioureas as neutral or monoanionic ligands to PdII and PtII through S and a pyrimidine-N. The IR spectra show shifts of CS and pyrimidine ring stretch bands to lower and higher frequencies, respectively. The 1H NMR spectra differentiate between H(4′) and H(6′) resonances and indicate downfield shifts for all protons of pyrimidine [H(4′), H(5′), and H(6′)], two resonances for two N‒H protons for complexes containing the neutral ligand (HL), and only one N‒H proton chemical shift for complexes containing the monoanion (L). 13C NMR chemical shifts of pyrimidine carbons are correlated with the type of bonding between PdII or PtII and pyrimidine-N. The magnetic susceptibilities suggest a diamagnetic planar structure for all complexes. Supplemental materials are available for this article. Go to the publisher's online edition of Phosphorus, Sulfur, and Silicon and the Related Elements to view the free supplemental file.

Characterizations of QF Rings

A ring R is called “left GP-injective” if for any 0 ≠ a ∈ R, there exists n > 0 such that a n ≠ 0 and a n R = r(l(a n )). It is proved that (1) every right Noetherian left GP-injective ring such that every complement left ideal is a left annihilator is a QF ring, (2) every left GP-injective ring with ACC on left annihilators such that every complement left ideal is a left annihilator is a QF ring, and (3) every left P-injective left CS ring satisfying ACC on essential right ideals is a QF ring. Several well-known results on QF rings are obtained as corollaries.

Sulfur‐containing mesoionic compounds: Theoretical study on structure and properties

AbstractThe structure of the mesoionic 1,3‐dithiolylium‐4‐olate (2) and thiopyrylium‐3‐olate (3) and of their derivatives were calculated by first‐principles methods. The compounds are considered as cylic derivatives of open‐chain thione S‐ylides. The extent of cyclic electron delocalization was discussed in terms of the molecular charge distribution and nucleus‐independent chemical shift values (NICs). The calculated CS bond of SCO of 2 was found to be exceedingly long (>1.80 Å). This result suggests a facile CS ring opening of 2 to S‐ketenyl dithiomethylester. The transition structure of this reaction is nearly isoenergetic with the product of the reaction. The calculated reaction barrier amounts to about 12 kcal/mol and the ring‐to‐chain valence isomerization should therefore occur under thermal conditions. This result is in line with the chemical reactivity of diphenyl‐2. According to time‐dependent density functional theory calculations (TD‐DFT) the sulfur‐containing mesoionic compounds 2 and 3 should be colored. The intense long wavelength absorption of diphenyl‐2 at about 550 nm is well reproduced. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

On the symmetry of the Goldie and CS conditions for prime rings

It is shown that: (a) If R R is a prime right Goldie right CS ring with right uniform dimension at least 2, then R R is left Goldie, left CS; (b) A semiprime ring R R is right Goldie left CS iff R R is left Goldie, right CS.

On CS Rings and QF Rings

In this paper, we shall discuss the conditions for a right SC right CS ring to be a QF ring. In particular, we prove that if R is a right SI right CS ring satisfying the reflexive orthogonal condition (*) and if every CS right R-module is Σ-CS, then R is a QF ring.

Correspondence Theorems for Nondegenerate Modules and their Endomorphism Rings

Let R U be a left R-module whose Morita context is nondegenerate and S = End(U) . We show the following: (1) There is a projectivity (that is, an order-preserving bijection) between the complement submodules of R U and the complement left ideals of S; (2) S is a left CS ring if and only if R U is a CS module; (3) S is a Baer and left CS ring if and only if R U is a nonsingular and CS module. Special cases include some earlier works.

A Lattice Isomorphism Theorem for Nonsingular Retractable Modules

AbstractLet RM be a nonsingular module such that B = EndR(M) is left nonsingular and has as its maximal left quotient ring, where is the injective hull of RM. Then it is shown that there is a lattice isomorphism between the lattice C(M) of all complement submodules of RM and the lattice C(B) of all complement left ideals of B, and that RM is a CS module if and only if B is a left CS ring. In particular, this is the case if RM is nonsingular and retractable.

Correspondence theorems for nondegenerate modules and their endomorphism rings

Let R U _RU be a left R-module whose Morita context is nondegenerate and S = End ( U ) S = {\text {End}}(U) . We show the following: (1) There is a projectivity (that is, an order-preserving bijection) between the complement submodules of R U _RU and the complement left ideals of S; (2) S is a left CS ring if and only if R U _RU is a CS module; (3) S is a Baer and left CS ring if and only if R U _RU is a nonsingular and CS module. Special cases include some earlier works.

Nonsingular retractable modules and their endomorphism rings

A module RM is said to be retractable if HomR (M, U) ≠ 0 for each nonzero submodule U of M. M is said to be a CS module if every complement submodule of M is a direct summand in M. Retractable modules are compared to nondegenerate modules on the one hand and to e–retractable modules on the other (nondegenerate implies retractable implies e–retractable); and it is shown that if M is nonsingular and retractable, then EndRM is a left CS ring if and only if M is a CS module.

Alkane rearrangement pathways on tellurium-based catalysts

Isotopic tracer studies and reaction pathway analyses suggest that olefins and diene and triene species are reactive intermediates in n-heptane dehydrocyclization on Te/NaX. Their concentration is limited by surface hydrogen overpressures caused by a rate-limiting hydrogen desorption step in the dehydrogenation sequence. The distribution of toluene isotopomers formed from dehydrocyclization of n-heptane-1- 13C is consistent with a reaction sequence involving thermal cyclization of an equilibrated mixture of conjugated and nonconjugated heptatrienes. Methylhexanes are formed predominantly by methyl and ethyl shift reactions of heptenes, and not by hydrogenolysis of CS ring species. Alkane hydrogenolysis on Te/NaX occurs predominantly by thermal cracking pathways of n-heptane and heptenes. No catalytic function for either (1,5) or (1,6) ring closure was observed on Te/Na X; a catalytic dehydrogenation function, however, suffices for dehydrocyclization to occur with high selectivity.

Spectroscopic studies of metal complexes containing π-delocalized sulphur ligands. The resonance-Raman spectra of the antitumor agent [Cu(kts)] and the related cation [Cu(H2kts)]2+

The resonance Raman spectra of the antitumor agent [3-ethoxy-2-oxobutyraldehyde bis(thiosemicarbazonato)-(2–)-N1N1′SS′]copper(II), [Cu(kts)], and the related cation [Cu(H2kts)]2+ have been measured. In the spectra of both compounds the bands most strongly enhanced are due to skeletal vibrations (CN and CS stretches and ring deformations), while band intensities of metal–ligand vibrations are only slightly modified. Excitation profiles show that the coupling of these vibrations with two π-delocalized sulphur-to-metal charge-transfer transitions is the mechanism responsible for Raman intensity enhancement.