In this paper we study M -small principally injective (in short, M -sp-injective) module which is the generalization of M -principally injective module. We prove that if M is finite dimensional and quasi-sp-injective then its endomorphism ring S is semi-local ring. We characterize the M -sp-injective module with the help of epi-retractable modules. Keywords: Small submodule; M -cyclic submodule; M -sp-injective modules; quasi-spinjective modules and epi-retractable module. AMS Subject Classification: 16D10, 16D50, 16D60 1250005-1 V. Kumar et al.N . A module M is called quasi-principally (or semi) injective, if it is M -principally injective. In this paper we introduce the notion of M -small principally injective modules and quasi-small principally injective modules which we abbreviate as Msp-injective and quasi-sp-injective modules. A submodule K of an R-module M is said to be small in M , written as K M , if for every submodule L ⊆ M withFor details see [1,5]. In this paper, we study M -sp-injective modules and M -sp-injective rings and also give an example of an M -sp-injective module which is not an M -principally injective module.Consider the following condition for an R-module M ., it is called continuous, if it satisfies (C 1 ) and (C 2 ), and quasi-continuous, if it satisfies (C 1 ) and (C 3 ). For undefined notations and terminologies see [1,14]. M-Small Principally Injective ModulesGeneralizing the notion of Sanh et al. [13], we now give a new definition. DefinitionAn R-module N is called M -small principally injective, if for every small M -cyclic submodule K of M , any homomorphism from K to N can be extended to a homomorphism from M to N . If M is a M -small principally injective module, then it is called a quasi-small principally injective module, or M -sp-injective module, and for short a ring R is called a right small-principally injective ring, if R R as a right R-module. Example(1) Every M -principally injective module is a M -sp-injective module.(2) Every semi-simple module is an M -sp-injective module.We now give an example of M -sp-injective modules which is not M -principally injective. ExampleZ is a Z-small principally injective module, but it is not Z-principally injective, because the only small Z-cyclic submodule of Z is 0.The following lemmas are the generalizations of [13, Lemmas 2.2-2.4]. Lemma 2.1. Let M i (1 ≤ i ≤ n) be M-sp-injective modules. Then n i=1 M i is M -sp-injective module. 1250005-2 M -SP-Injective Modules Proof. The proof is similar to that of [13, Lemma 2.2]. Lemma 2.2. Let X be an M-cyclic submodule of M . If X is M-sp-injective module, then it is a direct summand of M . Proof. The proof is similar to that of [13, Lemma 2.3]. Lemma 2.3. Every direct summand of M-sp-injective module is an M-sp-injective module. Proof. By the same argument as that given in [13, Lemma 2.4]. 1250005-5 V. Kumar et al. Proposition 2.6. For a hollow R-module M, the following conditions are equivalent :(1) M is quasi-sp-injective module;(2) M is quasi-principally injective module.Proof. It is ...
Background: The objective of the study was to evaluate the role of iron deficiency in febrile seizures.Methods: Case control study conducted at Pediatric department of Tertiary care hospital. A total 70 cases and 70 controls were included in the study. Consecutive cases and controls were selected. Cases were children of age group 6 months to 5 years with simple febrile seizures and controls were children of same age group with short febrile illness without any seizures. After consent, detailed history was taken and clinical examination were carried out for both groups. Blood investigations were done to diagnose iron deficiency in both cases and controls. Iron deficiency was diagnosed as per WHO criteria haemoglobin <11 g/dl in cases <5 years mean corpuscular volume<70 fl, mean corpuscular haemoglobin <27 pg and serum ferritin<30 microgram/dl.Results: Mean corpuscular volume was less than 70 fl/ml in 51% cases and 31% controls and mean corpuscular hemoglobin was less than 27 pg/ml in 84% cases and 40% of controls indicating statistically significant association of MCV and MCH between cases and controls .Serum ferritin was less than 30 ng/ml in 44% in cases as compared to 26% of controls, there was statistically significant difference between serum ferritin levels in cases and controls. From the above findings, study showed statistically significant association between iron deficiency anaemia and febrile seizures (p value <0.01; OR- 8.05 (3.6-17.93) (df-1).Conclusions: Iron deficiency is a major risk factor for simple febrile seizures in age group of 6 months to 60 months.
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Background: The objective of the study was to study the prevalence and various risk factors of dysnatremia in sick newborns admitted in neonatal intensive care unit (NICU).Methods: Cross sectional study conducted in Paediatric Department of tertiary care hospital from February 2016 to October 2016 which includes 384 neonates admitted to NICU during the study period. After informed consent, detailed history was taken and clinical examination carried out in both cases and controls. Blood investigations were done to diagnose sodium levels in sick neonates. Based on the corrected sodium values, the subjects were classified as having hyponatremia (serum sodium <135 meq/l), hypernatremia (serum sodium >145 meq/l) or normonatremia (serum sodium 135 to 145 meq/l).Results: The mean (SD) (range) serum sodium in sick newborns measured was 136.72 (6.7) (115-165) meq/l at a median (range) age of 56.97 (1-545) hours. Out of 384 sodium values obtained, 285 (74.2%) were sent on ≤3 days, 64 (16.7%) between 4th to the 6th day and 35 (9.1%) were sent on ≥7 days. The overall frequency of dysnatremia in 384 sodium values from 384 patients was 142 (37%). Hyponatremia was observed in 117 (30.5%) and hypernatremia in 25 (6.5%) of sodium values. Hyponatremia observed in term, low birth weight, very low birth weight and extremely low birth weight neonates were 16.4%, 25.2%, 67.2% and 100% respectively whereas hypernatremia were 10.1%, 4.6%, 3.4% and 0% respectively. Various risk factors for hyponatremia namely; prematurity, necrotizing enterocolitis, renal failure, birth asphyxia, sepsis, meningitis, vomiting/ nasogastric drainage.Conclusions: Hyponatremia are common in sick newborns in NICU.
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