Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities

The Levi theorem tells us that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. Therefore, it is important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this paper, we give a new connection between nilpotent Lie algebras and nilradicals of derivation Lie algebras of isolated complete intersection singularities. As an application, we obtain the correspondence between the nilpotent Lie algebras of dimension less than or equal to 7 and the nilradicals of derivation Lie algebras of isolated complete intersection singularities with modality less than or equal to 1. Moreover, we give a new characterization theorem for zero-dimensional simple complete intersection singularities.

New characterizations of S-coherent rings

In this paper, we introduce and study the class [Formula: see text]-[Formula: see text]-ML of [Formula: see text]-Mittag-Leffler modules with respect to all flat modules. We show that a ring [Formula: see text] is [Formula: see text]-coherent if and only if every ideal is in [Formula: see text]-[Formula: see text]-ML, if and only if [Formula: see text]-[Formula: see text]-ML is closed under submodules. As an application, we obtain the [Formula: see text]-version of Chase Theorem: a ring [Formula: see text] is [Formula: see text]-coherent if and only if any direct product of copies of [Formula: see text] is [Formula: see text]-flat, if and only if any direct product of flat [Formula: see text]-modules is [Formula: see text]-flat. Consequently, we provide an answer to the open question proposed by Bennis and El Hajoui [On [Formula: see text]-coherence, J. Korean Math. Soc. 55(6) (2018) 1499–1512].

Afterword

This short section reasserts and summarizes some of the key conclusions to the book, explaining how London responded to the strategic choices of Roman emperors and governors, initially as a gateway emporium and subsequently as a defended administrative enclave. Episodes of systematic change were also provoked by exogenous shock, and the effects of war and plague can be identified in the archaeological record from London. Imperial inputs helped London to recover from such events, but the city was wholly a creature of Rome and otherwise lacking in social capital. Its eventual failure was a direct product of the failure of the Roman administration. Some directions for future research are considered.

On Automorphisms of the Double Cover of a Circulant Graph

A graph $X$ is said to be unstable if the direct product $X \times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is nontrivially unstable if it is unstable, connected, and nonbipartite, and no two distinct vertices of $X$ have exactly the same neighbors. We find three new conditions that each imply a circulant graph is unstable. (These yield infinite families of nontrivially unstable circulant graphs that were not previously known.) We also find all of the nontrivially unstable circulant graphs of order $2p$, where $p$ is any prime number. Our results imply that there does not exist a nontrivially unstable circulant graph of order $n$ if and only if either $n$ is odd, or $n < 8$, or $n = 2p$, for some prime number $p$ that is congruent to $3$ modulo $4$.

A cost utility analysis alongside a cluster-randomised trial evaluating a minor ailment service compared to usual care in community pharmacy

Background Minor ailments are “self-limiting conditions which may be diagnosed and managed without a medical intervention”. A cluster randomised controlled trial (cRCT) was designed to evaluate the clinical, humanistic and economic outcomes of a Minor Ailment Service (MAS) in community pharmacy (CP) compared with usual care (UC). Methods The cRCT was conducted for 6 months from December 2017. The pharmacist-patient intervention consisted of a standardised face-to-face consultation on a web-based program using co-developed protocols, pharmacists’ training, practice change facilitators and patients’ educational material. Patients requesting a non-prescription medication (direct product request) or presenting minor ailments received MAS or UC and were followed-up by telephone 10-days after the consultation. The primary economic outcomes were incremental cost-utility ratio (ICUR) of the service and health related quality of life (HRQoL). Total costs included health system, CPs and patient direct costs: health professionals’ consultation time, medication costs, pharmacists’ training costs, investment of the pharmacy and consultation costs within the 10 days following the initial consultation. The HRQoL was obtained using the EuroQoL 5D-5L at the time of the consultation and at 10-days follow up. A sensitivity analysis was carried out using bootstrapping. There were two sub-group analyses undertaken, for symptom presentation and direct product requests, to evaluate possible differences. Results A total of 808 patients (323 MAS and 485 UC) were recruited in 27 CPs with 42 pharmacists (20 MAS and 22 UC). 64.7% (n = 523) of patients responded to follow-up after their consultation in CP. MAS patients gained an additional 0.0003 QALYs (p = 0.053). When considering only MAS patients presenting with symptoms, the ICUR was 24,733€/QALY with a 47.4% probability of cost-effectiveness (willingness to pay of 25,000€/QALY). Although when considering patients presenting for a direct product request, MAS was the dominant strategy with a 93.69% probability of cost-effectiveness. Conclusions Expanding community pharmacists’ scope through MAS may benefit health systems. To be fully cost effective, MAS should not only include consultations arising from symptom presentation but also include an oversight of self-selected products by patients. MAS increase patient safety through the appropriate use of non-prescription medication and through the direct referral of patients to GP. Trial registration ISRCTN, ISRCTN17235323. Registered 07/05/2021 - Retrospectively registered

On 1-absorbing δ-primary ideals

Let R be a commutative ring with nonzero identity. Let 𝒥(R) be the set of all ideals of R and let δ : 𝒥 (R) → 𝒥 (R) be a function. Then δ is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J ⊆ I, we have L ⊆ δ (L) and δ (J) ⊆ δ (I). Let δ be an expansion function of ideals of R. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of δ -primary ideals. A proper ideal I of R is said to be a 1-absorbing δ -primary ideal if whenever nonunit elements a, b, c ∈ R and abc ∈ I, then ab ∈ I or c ∈ δ (I). Moreover, we give some basic properties of this class of ideals and we study the 1-absorbing δ-primary ideals of the localization of rings, the direct product of rings and the trivial ring extensions.

Locally pro-p contraction groups are nilpotent

The authors have shown previously that every locally pro-p contraction group decomposes into the direct product of a p-adic analytic factor and a torsion factor. It has long been known that p-adic analytic contraction groups are nilpotent. We show here that the torsion factor is nilpotent too, and hence that every locally pro-p contraction group is nilpotent.

On rings whose quasi-injective modules are injective or semisimple

Two obvious classes of quasi-injective modules are those of semisimples and injectives. In this paper, we study rings with no quasi-injective modules other than semisimples and injectives. We prove that such rings fall into three classes of rings, namely, (i) QI-rings, (ii) rings with no middle class, or (iii) rings that decompose into a direct product of a semisimple Artinian ring and a strongly prime ring. Thus, we restrict our attention to only strongly prime rings and consider hereditary Noetherian prime rings to shed some light on this mysterious case. In particular, we prove that among these rings, QIS-rings which are not of type (i) or (ii) above are precisely those hereditary Noetherian prime rings which are idealizer rings from non-simple QI-overrings.