Weak Global Dimension of Coherent Rings

In this article, we study the weak global dimension of coherent rings in terms of the left FP-injective resolutions of modules. Let R be a left coherent ring and ? ? the class of all FP-injective left R-modules. It is shown that wD(R) ≤ n (n ≥ 1) if and only if every nth ? ?-syzygy of a left R-module is FP-injective; and wD(R) ≤ n (n ≥ 2) if and only if every (n ? 2)th ? ?-syzygy in a minimal ? ?-resolution of a left R-module has an FP-injective cover with the unique mapping property. Some results for the weak global dimension of commutative coherent rings are also given.     

Relative Flatness,Mittag–Leffler Modules,and Endocoherence

Let M _R be a right R-module over a ring R with S = End(M _R ). We study the coherence of the left S-module _S M relative to a hereditary torsion theory for the category of right R-modules. Various results are developed, many extending known results.     

On Divisible and Torsionfree Modules

A ring R is called left P-coherent in case each principal left ideal of R is finitely presented. A left R-module M (resp. right R-module N) is called D-injective (resp. D-flat) if Ext¹(G, M) = 0 (resp. Tor₁(N, G) = 0) for every divisible left R-module G. It is shown that every left R-module over a left P-coherent ring R has a divisible cover; a left R-module M is D-injective if and only if M is the kernel of a divisible precover A → B with A injective; a finitely presented right R-module L over a left P-coherent ring R is D-flat if and only if L is the cokernel of a torsionfree preenvelope K → F with F flat. We also study the divisible and torsionfree dimensions of modules and rings. As applications, some new characterizations of von Neumann regular rings and PP rings are given.     

The connes spectrums of certain finite non-abelian groups

In this article, we investigate when every simple module has a projective (pre)envelope. It is proven that (1) every simple right R-module has a projective preenvelope if and only if the left annihilator of every maximal right ideal of R is finitely generated; (2) every simple right R-module has an epic projective envelope if and only if R is a right PS ring; (3) Every simple right R-module has a monic projective preenvelope if and only if R is a right Kasch ring and the left annihilator of every maximal right ideal of R is finitely generated.     

Chemo‐ and Regioselective Dihydroxylation of Benzene to Hydroquinone Enabled by Engineered Cytochrome P450 Monooxygenase

Hydroquinone (HQ) is produced commercially from benzene by multi‐step Hock‐type processes with equivalent amounts of acetone as side‐product. We describe an efficient biocatalytic alternative using the cytochrome P450‐BM3 monooxygenase. Since the wildtype enzyme does not accept benzene, a semi‐rational protein engineering strategy was developed. Highly active mutants were obtained which transform benzene in a one‐pot sequence first into phenol and then regioselectively into HQ without any overoxidation. A computational study shows that the chemoselective oxidation of phenol by the P450‐BM3 variant A82F/A328F leads to the regioselective formation of an epoxide intermediate at the C3=C4 double bond, which departs from the binding pocket and then undergoes fragmentation in aqueous medium with exclusive formation of HQ. As a practical application, an E. coli designer cell system was constructed, which enables the cascade transformation of benzene into the natural product arbutin, which has anti‐inflammatory and anti‐bacterial activities.     

Suppressed Mobility of Negative Charges in Polymer Electrolytes with an Ether‐Functionalized Anion

Suppressing the mobility of anionic species in polymer electrolytes (PEs) is essential for mitigating the concentration gradient and internal cell polarization, and thereby improving the stability and cycle life of rechargeable alkali metal batteries. Now, an ether‐functionalized anion (EFA) is used as a counter‐charge in a lithium salt. As the salt component in PEs, it achieves low anionic diffusivity but sufficient Li‐ion conductivity. The ethylene oxide unit in EFA endows nanosized self‐agglomeration of anions and trapping interactions between the anions and its structurally homologous matrix, poly(ethylene oxide), thus suppressing the mobility of negative charges. In contrast to previous strategies of using anion traps or tethering anions to a polymer/inorganic backbone, this work offers a facile and elegant methodology on accessing selective and efficient Li‐ion transport in PEs and related electrolyte materials (for example, composites and hybrid electrolytes).     

Stereoconvergent,Redox‐Neutral Access to Tetrahydroquinoxalines through Relay Epoxide Opening/Amination of Alcohols

We present an economical catalytic procedure to convert readily available 1,2‐diaminobenzenes and terminal epoxides into valuable 1,2,3,4‐tetrahydroquinoxalines in a highly enantioselective fashion. This procedure operates through relay zinc and iridium catalysis, and achieves redox‐neutral and stereoconvergent production of valuable chiral heterocycles from racemic starting materials with water as the only side product. The use of commercially available reagents and catalysts and a convenient procedure also make this catalytic method attractive for practical application.