Finite groups of units of finite characteristic rings

In \cite[Problem 72]{Fuchs60} Fuchs asked the following question: which groups can be the group of units of a commutative ring? In the following years, some partial answers have been given to this question in particular cases. The aim of the present paper is to address Fuchs' question when $A$ is a {\it finite characteristic ring}. The result is a pretty good description of the groups which can occur as group of units in this case, equipped with examples showing that there are obstacles to a "short" complete classification. As a byproduct, we are able to classify all possible cardinalities of the group of units of a finite characteristic ring, so to answer Ditor's question \cite{ditor}

On wild extensions of a p-adic field

In this paper we consider the problem of classifying the isomorphism classes of extensions of degree pk of a p-adic field, restricting to the case of extensions without intermediate fields. We establish a correspondence between the isomorphism classes of these extensions and some Kummer extensions of a suitable field F containing K. We then describe such classes in terms of the representations of Gal(F/K). Finally, for k = 2 and for each possible Galois group G, we count the number of isomorphism classes of the extensions whose normal closure has a Galois group isomorphic to G. As a byproduct, we get the total number of isomorphism classes

A note on upper ramification jumps in Abelian extensions of exponent p

In this paper we present a classification of the possible upper ramification jumps for an elementary Abelian p-extension of ap-adic field. The fundamental step for the proof of the main result is the computation of the ramification filtration for the maximal elementary Abelian p-extension of the base field K. This result generalizes [3, Lemma 9, p. 2861, where the same result is proved under the assumption that K contains a primitive p-th root of unity. To deal with this general case we use class field theory and the explicit relations between the normic group of an extension and its ramification jumps, and we obtain necessary and sufficient conditions for the upper ramification jumps of an elementary Abelian p-extension of K

Hopf-Galois structures on extensions of degree p2qp^{2} q and skew braces of order p2qp^{2} q: the elementary abelian Sylow pp-subgroup case

Let $p, q$ be distinct primes, with $p > 2$. In a previous paper we classified the Hopf-Galois structures on Galois extensions of degree $p^{2} q$, when the Sylow $p$-subgroups of the Galois group are cyclic. This is equivalent to classifying the skew braces of order $p^2q$, for which the Sylow $p$-subgroups of the multiplicative group is cyclic. In this paper we complete the classification by dealing with the case when the Sylow $p$-subgroups of the Galois group are elementary abelian. According to Greither and Pareigis, and Byott, we will do this by classifying the regular subgroups of the holomorphs of the groups $(G, \cdot)$ of order $p^{2} q$, in the case when the Sylow $p$-subgroups of $G$ are elementary abelian. We rely on the use of certain gamma functions $\gamma:G\to \operatorname{Aut}(G)$. These functions are in one-to-one correspondence with the regular subgroups of the holomorph of $G$, and are characterised by the functional equation $\gamma(g^{\gamma(h)} \cdot h) = \gamma(g) \gamma(h)$, for $g, h \in G$. We develop methods to deal with these functions, with the aim of making their enumeration easier and more conceptual.Comment: 95 page

Hopf-Galois structures on extensions of degree p2qp^{2} q and skew braces of order p2qp^{2} q: the cyclic Sylow pp-subgroup case

\DeclareMathOperator{\Aut}{Aut}Let $p, q$ be distinct primes, with $p > 2$. We classify the Hopf-Galois structures on Galois extensions of degree $p^{2} q$, such that the Sylow $p$-subgroups of the Galois group are cyclic. This we do, according to Greither and Pareigis, and Byott, by classifying the regular subgroups of the holomorphs of the groups $(G, \cdot)$ of order $p^{2} q$, in the case when the Sylow $p$-subgroups of $G$ are cyclic. This is equivalent to classifying the skew braces $(G, \cdot, \circ)$. Furthermore, we prove that if $G$ and $\Gamma$ are groups of order $p^{2} q$ with non-isomorphic Sylow $p$-subgroups, then there are no regular subgroups of the holomorph of $G$ which are isomorphic to $\Gamma$. Equivalently, a Galois extension with Galois group $\Gamma$ has no Hopf-Galois structures of type $G$. Our method relies on the alternate brace operation $\circ$ on $G$, which we use mainly indirectly, that is, in terms of the functions \gamma : G \to \Aut(G) defined by $g \mapsto (x \mapsto (x \circ g) \cdot g^{-1})$. These functions are in one-to-one correspondence with the regular subgroups of the holomorph of $G$, and are characterised by the functional equation $\gamma(g^{\gamma(h)} \cdot h) = \gamma(g) \gamma(h)$, for $g, h \in G$. We develop methods to deal with these functions, with the aim of making their enumeration easier, and more conceptual.Comment: 43 page

Serre's "formule de masse" in prime degree

For a local field F with finite residue field of characteristic p, we describe completely the structure of the filtered F_p[G]-module K^*/K^*p in characteristic 0 and $K^+/\wp(K^+) in characteristic p, where K=F(\root{p-1}\of F^*) and G=\Gal(K|F). As an application, we give an elementary proof of Serre's mass formula in degree p. We also determine the compositum C of all degree p separable extensions with solvable galoisian closure over an arbitrary base field, and show that C is K(\root p\of K^*) or K(\wp^{-1}(K)) respectively, in the case of the local field F. Our method allows us to compute the contribution of each character G\to\F_p^* to the degree p mass formula, and, for any given group \Gamma, the contribution of those degree p separable extensions of F whose galoisian closure has group \Gamma.Comment: 36 pages; most of the new material has been moved to the new Section

L-Idose: an attractive substrate alternative to d-glucose for measuring aldose reductase activity

Although glucose is one of the most important physio-pathological substrates of aldose reductase, it is not an easy molecule for in vitro investigation into the enzyme. In many cases alternative aldoses have been used for kinetic characterization and inhibition studies. However these molecules do not completely match the structural features of glucose, thus possibly leading to results that are not fully applicable to glucose. We show how aldose reductase is able to act efficiently on L-idose, the C-5 epimer of D-glucose. This is verified using both the bovine lens and the human recombinant enzymes. While the kcat values obtained are essentially identical to those measured for D-glucose, a significant decrease in KM was observed. This can be due to the significantly higher level of the free aldehyde form present in L-idose compared to D-glucose. We believe that L-idose is the best alternative to D-glucose in studies on aldose reductase

In search for multi-target ligands as potential agents for diabetes mellitus and its complications—a structure-activity relationship study on inhibitors of aldose reductase and protein tyrosine phosphatase 1b

Diabetes mellitus (DM) is a complex disease which currently affects more than 460 million people and is one of the leading cause of death worldwide. Its development implies numerous metabolic dysfunctions and the onset of hyperglycaemia-induced chronic complications. Multiple ligands can be rationally designed for the treatment of multifactorial diseases, such as DM, with the precise aim of simultaneously controlling multiple pathogenic mechanisms related to the disease and providing a more effective and safer therapeutic treatment compared to combinations of selective drugs. Starting from our previous findings that highlighted the possibility to target both aldose reductase (AR) and protein tyrosine phosphatase 1B (PTP1B), two enzymes strictly implicated in the development of DM and its complications, we synthesised 3-(5-arylidene-4-oxothiazolidin-3-yl)propanoic acids and analogous 2-butenoic acid derivatives, with the aim of balancing the effectiveness of dual AR/PTP1B inhibitors which we had identified as designed multiple ligands (DMLs). Out of the tested compounds, 4f exhibited well-balanced AR/PTP1B inhibitory effects at low micromolar concentrations, along with interesting insulin-sensitizing activity in murine C2C12 cell cultures. The SARs here highlighted along with their rationalization by in silico docking experiments into both target enzymes provide further insights into this class of inhibitors for their development as potential DML antidiabetic candidates

Specifically targeted modification of human aldose reductase by physiological disulfides

Aldose reductase is inactivated by physiological disulfides such as GSSG and cystine. To study the mechanism of disulfide-induced enzyme inactivation, we examined the rate and extent of enzyme inactivation using wild-type human aldose reductase and mutants containing cysteine-to-serine substitutions at positions 80 (C80S), 298 (C298S), and 303 (C303S). The wild-type, C80S, and C303S enzymes lost >80% activity following incubation with GSSG, whereas the C298S mutant was not affected. Loss of activity correlated with enzyme thiolation. The binary enzyme-NADP+ complex was less susceptible to enzyme thiolation than the apoenzyme. These results suggest that thiolation of human aldose reductase occurs predominantly at Cys-298. Energy minimization of a hypothetical enzyme complex modified by glutathione at Cys-298 revealed that the glycyl carboxylate of glutathione may participate in a charged interaction with His-110 in a manner strikingly similar to that involving the carboxylate group of the potent aldose reductase inhibitor Zopolrestat. In contrast to what was observed with GSSG and cystine, cystamine inactivated the wild-type enzyme as well as all three cysteine mutants. This suggests that cystamine-induced inactivation of aldose reductase does not involve modification of cysteines exclusively at position 80, 298, or 303