Solving an Open Problem of K\"othe and its Dual Problem on Non-Commutative Rings

We solve the classical K\"othe's Problem, concerning the structure of non-commutative rings with the property that: "every left module is a direct sum of cyclic modules". A ring $R$ is called a ${\it left~K\"othe~ring}$ if every left $R$-module is a direct sum of cyclic $R$-modules, and a ring $R$ is called a ${\it left~Kawada~ring}$ if any ring Morita equivalent to $R$ is a left K\"othe ring. In 1934 K\"othe showed that all Artinian principal ideal rings are left K\"othe rings. In 1951 Cohen and Kaplansky proved that all commutative K\"othe rings are Artinian principal ideal rings. In 1962-1965, Kawada completely solved K\"othe's Problem for basic finite dimensional algebras. So far, the problem was still open in the non-commutative setting. In this paper, we break the class of left K\"othe rings into three categories of nested: ${\it left~K\"othe~rings}$, ${\it strongly~left~K\"othe~rings}$ and ${\it very~strongly~left~K\"othe~rings}$, and then, we solve the K\"othe's Problem through the several characterizations and properties that we provide for all of them. Also as an application, we introduce and describe the dual of the above as: ${\it left~co}$-${\it K\"othe~rings}$, ${\it strongly~left~co}$-${\it K\"othe~rings}$, and ${\it very~strongly~left~co}$-${\it K\"othe~rings}$. Finally, we present several characterizations of left Kawada rings, which generalizes a well-known result of Ringel on finite dimensional Kawada algebras.Comment: The new version of the paper is 45 pages, and there were some errors in the previous versions that have been fixed in this version. Therefore, this version is the most complete version and has validit

Left Co-K\"othe Rings and Their Characterizations

The classical K\"othe's problem posed by G. K\"othe in 1935 asks to describe the rings $R$ such that every left $R$-module is a direct sum of cyclic modules (are known as left K\"othe rings). K\"othe, Cohen and Kaplansky solved this problem for all commutative rings (that are Artinian principal ideal rings). During the years 1962 to 1965, Kawada solved the K\"othe's problem for basic fnite-dimensional algebras. But, so far, the K\"othe's problem was open in the non-commutative setting. Recently, in the paper ["Several Characterizations of Left K\"othe Rings", submitted], we brook the class of left K\"othe rings into three categories of nested: left K\"othe rings, strongly left K\"othe rings and very strongly left K\"othe rings, and then, we solved the K\"othe's problem by giving several characterizations of these rings in terms of describing the indecomposable modules. In this paper, we will introduce the Morita duality of these notions as co-K\"othe rings, left co-K\"othe rings and strongly left co-K\"othe rings, and then, we give several structural characterizations for each of them.Comment: This paper is a sequel of the paper entitled "Several Characterizations of Left K\"othe Rings" (see arXiv:2206.06453

The effect of Directed medical play on the severity of pain during burn dressing change: Clinical Randomized trial

Introduction: Pain management in children with burn is one of the most important issues in pediatric medicine. Non-pharmacological methods of pain control can play an important role in patients. Therefore this study was conducted to investigate the effect of medical directed play on the severity of pain during burn dressing change. Methods: To conduct this two-group clinical trial, 78 children of 3-6 years old with burn surface area ranging less than 25% who had referred to Shahid Motahari Medical Center for their first burn dressing change, were selected. After obtaining the ethics approval for conducting the study, the patients were placed as random allocation in control and intervention groups. For the intervention group, the pre-dressed directed medical play was done and routine care was taken for the control group. The FLACC scale was used in both groups to measure the severity of pain before and during the dressing; the arterial blood oxygen level and heart rate were also measured and compared. For data analysis, SPSS 20 was used.  Results: There were significant differences between the two groups during burn dressing change of mean pain (P=0.041) and pulse (P=0.037), and the mean of both indices in the control group was high. The mean of the arterial blood oxygen level during burn dressing change did not differ significantly between the two groups (P=1). Conclusion: Directed Medical play effectively reduces the pain of pediatric burn dressing change. This technique is cost-effective, so it can be widely used for pain management in pediatric department

Vanillic acid protects mortality and toxicity induced by N-ethyl-N-nitrosourea in mice; in vivo model of chronic lymphocytic leukemia

Alkylating agents such as N-Ethyl-N-Nitrosourea (ENU) are ubiquitous within living cells and in the environment. This study designed to evaluate the chemopreventive activity of vanillic acid on ENU-induced toxicity and carcinogenesis in mice as an animal model of chronic lymphocytic leukemia (CLL). The female, Swiss albino mice were divided into three groups each with 7 mice, group I received normal saline, group II, mice received ENU at a dose of 80 mg/kg body weight i.p. to induce CLL on the 31th day of the study, and group III, the mice pretreated with vanillic acid at a dose of 20 mg/kg body weight/day, i.p. up to 30 days and received ENU. The animals were monitored for weight changes and mortality during 120 days, and then were sacrificed for isolation of lymphocytes, as target cells in CLL. Cellular parameters like reactive oxygen species (ROS) formation, malondialdehyde (MDA) production, depletion of glutathione (GSH), mitochondrial membrane potential (MMP) and lysosomal membrane integrity were studied. We found that pretreatment with vanillic acid significantly increased the survival of mice up to 57%, delay in death time (30%) and prevented weight changes after exposure to ENU. In addition, it was found that vanillic acid protected ROS formation, lipid peroxidation mitochondrial dysfunction, and lysosomal membrane destabilization in isolated lymphocytes. These data suggest that vanillic acid exhibited significant protection against ENU-induced toxicity and carcinogenicity, which might be related to the protection of the mitochondria and lysosomes and the reduction of ROS formation and oxidative stress