Let R be a commutative ring with identity 1 ¹ 0, and let M be a unitary left module over R. A submodule N of an R-module M is called essential, if whenever N ⋂ L = (0), then L = (0) for every submodule L of M. In this case, we write N ≤e M. An R-module M is called extending, if every submodule of M is an essential in a direct summand of M. A submodule N of an R-module M is called semi-essential (denoted by N ≤sem M), if N ∩ P ≠ (0) for each nonzero prime submodule P of M. The main purpose of this work is to determine and study two new concepts (up to our knowledge) which are St-closed submodules and semi-extending modules. St-closed submodules is contained properly in the class of closed submodules, where a submodule N of M is called St-closed in M, if N has no proper semi-essential extension in M, i.e if there exists a submodule K of M such that N is a semi-essential submodule of K, then N = K. We investigate the main properties of this type of submodules, and discuss some results that are useful in our work. The class of semi-extending modules is a generalization to the notion of extending modules, where an R-module M is called semi-extending, if every submodule of M is a semi-essential in a direct summand of M. Various properties of semi-extending modules are obtained, and we study the relationships between this class of modules and other related concepts.
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The article deals with the role of metaphors in forming the plot of L. Ulitskaya’s family chronicle “Medea and Her Children”. The author of the article describes the results of the next stage of research related to the works of Lyudmila Evgenievna Ulitskaya, a representative of modern Russian prose. The analysis of tropes and figures in the works written at the turn of the XXth – XXIth centuries is of importance for the study of the modern state of Russian language as an independent system. “Medea and Her Children” is one of the works by L. Ulitskaya (written in 1996), which, like her other works, is characterized by a unique style of narration, rich in vocabulary, lexical, semantic and stylistic diversity of the author’s word
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Odontogenic cysts and tumors often form hard and soft structures that resemble odontogenesis. It is well known that amyloid is produced in Pindborg tumors; however, it is still debatable whether it is also formed in other odontogenic tumors and cysts. This study aimed to detect the presence of amyloid in different odontogenic cysts and tumors in correlation to matrix proteins secreted during enamel formation; namely amelogenin and odontogenic ameloblast‐associated protein.
This study included formalin fixed paraffin embedded tissue blocks of 106 different types of odontogenic
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Two different approaches, univariate and multivariate (simplex method), have been used to obtain the optimum conditions for the quantitative Spectrophotometric determination of Eu3+ using Solochrome violet RS (3-Hydroxy-4-(2-hydroxy phenyl azo) naphthalene -1sulfonic acid) (SVRS) as a chromogenic reagent. The investigation shows that Eu3+ ion forms a wine-red complex with SVRS in alkaline buffer solution having a maximum absorbance at 464 nm against reagent blank. Calibration graphs obtained under univariate and simplex were found to be linear in the range of (0.30-8.0) µg/ml with detection limit 0.061µg/ml and molar absorptivity of 9877.66 L/mol.cm and (0.40-10.0)µg/ml with
... Show MoreIntroduction to Medical and Biological Statistics for Pharmacy Students and Medical Groups (Undergraduate & Postgraduate) - ISBNiraq.org
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