Left dihedral codes over finite chain rings

Let R be a finite commutative chain ring, $D_{2n}$ be the dihedral group of size 2n and $R[D_{2n}]$ be the dihedral group ring. In this paper, we completely characterize left ideals of $R[D_{2n}]$ (called left $D_{2n}$-codes) when $\gcd (char(R),n)=1$. In this way, we explore the structure of some skew-cyclic codes of length 2 over R and also over $R\times S$, where S is an isomorphic copy of R. As a particular result, we give the structure of cyclic codes of length 2 over R. In the case where $R={\mathbb{F}}_{p^m}$ is a Galois field, we give a classification for left $D_{2N}$-codes over ${\mathbb{F}}_{p^m}$, for any positive integer N. In both cases we determine dual codes and identify self-dual ones.     

Shimalactone Biosynthesis Involves Spontaneous Double Bicyclo-Ring Formation with 8π-6π Electrocyclization

Shimalactones A and B are neuritogenic polyketides possessing characteristic oxabicyclo[2.2.1]heptane and bicyclo[4.2.0]octadiene ring systems that are produced by the marine fungus Emericella variecolor GF10. We identified a candidate biosynthetic gene cluster and conducted heterologous expression analysis. Expression of ShmA polyketide synthase in Aspergillus oryzae resulted in the production of preshimalactone. Aspergillus oryzae and Saccharomyces cerevisiae transformants expressing ShmA and ShmB produced shimalactones A and B, thus suggesting that the double bicyclo-ring formation reactions proceed non-enzymatically from preshimalactone epoxide. DFT calculations strongly support the idea that oxabicyclo-ring formation and 8π-6π electrocyclization proceed spontaneously after opening of the preshimalactone epoxide ring through protonation. We confirmed the formation of preshimalactone epoxide in vitro, followed by its non-enzymatic conversion to shimalactones in the dark.     

Dimerisation of Dipiperidinoacetylene: Convenient Access to Tetraamino-1,3-Cyclobutadiene and Tetraamino-1,2-Cyclobutadiene Metal Complexes

The reaction of 1,2-dipiperidinoacetylene ( 1 ) with 0.5 equivalents of SnCl₂ or GeCl₂⋅dioxane afforded the 1,2,3,4-tetrapiperidino-1,3-cyclobutadiene tin and germanium dichloride complexes 2 a and 2 b , respectively. A competing redox reaction was observed with excess amounts of SnCl₂, which produced a tetrapiperidinocyclobutadiene dication with two trichlorostannate(II) counterions. Heating neat 1 to 110 °C for 16 h cleanly produced the dimer 1,3,4,4-tetrapiperidino-3-buten-1-yne ( 3 ); its reaction with stoichiometric amounts of SnCl₂ or GeCl₂⋅dioxane furnished the 1,3,4,4-tetrapiperidino-1,2-cyclobutadiene tin and germanium dichloride complexes 4 a and 4 b , respectively. Transition-metal complexes containing this novel four-membered cyclic bent allene (CBA) ligand were prepared by reaction of 3 with [(tht)AuCl], [RhCl(CO)₂]₂, and [(Me₃N)W(CO)₅] to form [(CBA)AuCl] ( 5 ), [(CBA)RhCl(CO)₂] ( 6 ), and [(CBA)W(CO)₅] ( 7 ). The molecular structures of all compounds 2 – 7 were determined by X-ray diffraction analyses, and density functional theory (DFT) calculations were carried out to rationalise the formation of 3 and 4 a .     

The cancellation property of rings

A ring R is said to have the finitely generated cancellation property provided that the module isomorphism R⊕B?R⊕C implies B?C for any finitely generated R-modules B and C. It is proved that R has this property is equivalent to the existence of the cancellation matrices over R. Moreover, the structure of such matrices is investigated and finite weakly stable rings are characterized in terms of their cancellation matrices.     

On graded nil clean rings

In this paper we introduce and study the notion of a graded (strongly) nil clean ring which is group graded. We also deal with extensions of graded (strongly) nil clean rings to graded matrix rings and to graded group rings. The question of when nil cleanness of the component, which corresponds to the neutral element of a group, implies graded nil cleanness of the whole graded ring is examined. Similar question is discussed in the case of groupoid graded rings as well.     

Quasi-ordered rings - A uniform approach to orderings and valuations

A quasi-order on a set S is a binary, reflexive and transitive relation on S. In [3 Fakhruddin, S. M. (1987). Quasi-ordered fields. J. Pure Appl. Algebra 45:207–210.[Crossref], [Web of Science ®] , [Google Scholar]], Fakhruddin introduced the notion of (totally) quasi-ordered fields and showed that each such field is either an ordered field or else a valued field. Hence, quasi-ordered fields are very well suited to treat ordered and valued fields simultaneously. The aim of the present paper is to prove that an analogous dichotomy holds for commutative rings with 1 as well.