Sums of units in function fields II - The extension problem

In 2007, Jarden and Narkiewicz raised the following question: Is it true that each algebraic number field has a finite extension L such that the ring of integers of L is generated by its units (as a ring)? In this article, we answer the analogous question in the function field case. More precisely, it is shown that for every finite non-empty set S of places of an algebraic function field F | K over a perfect field K, there exists a finite extension F' | F, such that the integral closure of the ring of S-integers of F in F' is generated by its units (as a ring).Comment: 12 page

Minimal Reversible Nonsymmetric Rings

Marks showed that $\mathbb{F}_2Q_8$, the $\mathbb{F}_2$ group algebra over the quaternion group, is a reversible nonsymmetric ring, then questioned whether or not this ring is minimal with respect to cardinality. In this work, it is shown that the cardinality of a minimal reversible nonsymmetric ring is indeed 256. Furthermore, it is shown that although $\mathbb{F}_2Q_8$ is a duo ring, there are also examples of minimal reversible nonsymmetric rings which are nonduo

6-(4-Fluorophenyl)-8-phenyl-2,3-dihydro-4H-imidazo[5,1-b][1,3]thiazin-4-one: an unusual [6-5] fused-ring system

The title compound, C₁₈H₁₃FN₂OS, is the first structural example of a [6-5] fused ring incorporating the 2,3-dihydro-4H-imidazo[5,1-b][1,3]thiazin-4-one molecular scaffold. The six-membered 2,3-dihydro-1,3-thiazin-4-one ring adopts an envelope conformation, with the S-CH₂ C atom displaced by 0.761 (2) Å from the five-atom plane (all within 0.05 Å of the mean plane). The imidazole ring is planar. The phenyl ring is twisted from coplanarity with the imidazole ring by 23.84 (5)° and the 4-fluorophenyl ring is twisted by 53.36 (6)°, due to a close C(aryl)-H...O=C contact with the thiazin-4-one carbonyl O atom. The primary intermolecular interaction involves a CH₂ group with the F atom [C...F = 3.256 (2) Å and C-H...F = 137°]

The F-signature and strong F-regularity

We show that the F-signature of a local ring of characteristic p, defined by Huneke and Leuschke, is positive if and only if the ring is strongly F-regular.Comment: revised version, incorporating referee's comments. 6 page