KERNELS OF MORPHISMS BETWEEN INDECOMPOSABLE INJECTIVE MODULES

AbstractWe show that the endomorphism rings of kernels ker ϕ of non-injective morphisms ϕ between indecomposable injective modules are either local or have two maximal ideals, the module ker ϕ is determined up to isomorphism by two invariants called monogeny class and upper part, and a weak form of the Krull–Schmidt theorem holds for direct sums of these kernels. We prove with an example that our pathological decompositions actually take place. We show that a direct sum ofnkernels of morphisms between injective indecomposable modules can have exactlyn! pairwise non-isomorphic direct-sum decompositions into kernels of morphisms of the same type. IfERis an injective indecomposable module andSis its endomorphism ring, the duality Hom(−,ER) transforms kernels of morphismsER→ERinto cyclically presented left modules over the local ringS, sending the monogeny class into the epigeny class and the upper part into the lower part

Preventing microbial infections with natural phenolic compounds

The struggle between humans and pathogens has taken and is continuing to take countless lives every year. As the misusage of conventional antibiotics increases, the complexity associated with the resistance mechanisms of pathogens has been evolving into gradually more clever mechanisms, diminishing the effectiveness of antibiotics. Hence, there is a growing interest in discovering novel and reliable therapeutics able to struggle with the infection, circumvent the resistance and defend the natural microbiome. In this regard, nature-derived phenolic compounds are gaining considerable attention due to their potential safety and therapeutic effect. Phenolic compounds comprise numerous and widely distributed groups with different biological activities attributed mainly to their structure. Investigations have revealed that phenolic compounds from natural sources exhibit potent antimicrobial activity against various clinically relevant pathogens associated with microbial infection and sensitize multi-drug resistance strains to bactericidal or bacteriostatic antibiotics. This review outlines the current knowledge about the antimicrobial activity of phenolic compounds from various natural sources, with a particular focus on the structure-activity relationship and mechanisms of actions of each class of natural phenolic compounds, including simple phenols, phenolic acids, coumarin, flavonoids, tannins, stilbenes, lignans, quinones, and curcuminoids.The authors acknowledge the financial support by the Portuguese Foundation for Science and Technology (FCT) through the doctoral grant and junior research contract with the reference number PD/BD/150521/2019 (K.E.) and CEECIND/01026/2018 (J.M.S.), respectively

KERNELS OF MORPHISMS BETWEEN INDECOMPOSABLE INJECTIVE MODULES

AbstractWe show that the endomorphism rings of kernels ker ϕ of non-injective morphisms ϕ between indecomposable injective modules are either local or have two maximal ideals, the module ker ϕ is determined up to isomorphism by two invariants called monogeny class and upper part, and a weak form of the Krull–Schmidt theorem holds for direct sums of these kernels. We prove with an example that our pathological decompositions actually take place. We show that a direct sum ofnkernels of morphisms between injective indecomposable modules can have exactlyn! pairwise non-isomorphic direct-sum decompositions into kernels of morphisms of the same type. IfERis an injective indecomposable module andSis its endomorphism ring, the duality Hom(−,ER) transforms kernels of morphismsER→ERinto cyclically presented left modules over the local ringS, sending the monogeny class into the epigeny class and the upper part into the lower part

DIRECT SUMS OF INFINITELY MANY KERNELS

AbstractLet be the class of all rightR-modules that are kernels of nonzero homomorphisms φ:E1→E2for some pair of indecomposable injective rightR-modulesE1,E2. In a previous paper, we completely characterized when two direct sumsA1⊕⋯⊕AnandB1⊕⋯⊕Bmof finitely many modulesAiand Bjin are isomorphic. Here we consider the case in which there are arbitrarily, possibly infinitely, manyAiandBjin . In both the finite and the infinite case, the behaviour is very similar to that which occurs if we substitute the class with the class of all uniserial rightR-modules (a module is uniserial when its lattice of submodules is linearly ordered)

Estimating the Impact of the Recent Economic Crisis on Work Time in Turkey

This paper provides estimates of the impact of the recent economic crisis on paid and unpaid work time in Turkey. The data used in this study come from the first and only time-use survey available at the national level. Infrequency of collection of time-use data in Turkey does not allow us to make a direct comparison of pre-versus postcrisis time-use patterns. We introduce a tractable way for estimating these possible effects by measuring the impact of an increase in unemployment risk on time-use patterns of women and men living in couple households. The method developed here can be applied to other developing-country cases where there is a lack of longitudinal data availability. Our findings support the argument that economic crises reinforce the preexisting gender inequalities in work time

International Consensus Statement on Rhinology and Allergy: Rhinosinusitis

Background: The 5 years since the publication of the first International Consensus Statement on Allergy and Rhinology: Rhinosinusitis (ICAR‐RS) has witnessed foundational progress in our understanding and treatment of rhinologic disease. These advances are reflected within the more than 40 new topics covered within the ICAR‐RS‐2021 as well as updates to the original 140 topics. This executive summary consolidates the evidence‐based findings of the document. Methods: ICAR‐RS presents over 180 topics in the forms of evidence‐based reviews with recommendations (EBRRs), evidence‐based reviews, and literature reviews. The highest grade structured recommendations of the EBRR sections are summarized in this executive summary. Results: ICAR‐RS‐2021 covers 22 topics regarding the medical management of RS, which are grade A/B and are presented in the executive summary. Additionally, 4 topics regarding the surgical management of RS are grade A/B and are presented in the executive summary. Finally, a comprehensive evidence‐based management algorithm is provided. Conclusion: This ICAR‐RS‐2021 executive summary provides a compilation of the evidence‐based recommendations for medical and surgical treatment of the most common forms of RS