### About the Closed Quasi Injective S-Acts Over Monoids

Received: 8 September 2019     Accepted: 15 October 2019     Published: 24 October 2019
Abstract

The aim of introducing and studying the notion of closed quasi injective S-act is to create a basis facilitate for the exchange ideas between module theory and act theory. As well as it represents a generalization of the quasi-injective act. The quasi-injective act was first introduced and studied by A. M. Lopez, Jr. and J. K. Luedeman, 1979. Then the author was one of the researchers which introduced several generalizations for this notion from several aspects because of its importance. More accurately, the contribution of this paper to the field of competence can be summarized into three points as follows: First: The possibilities for applying the topic of this article helps researchers about how can connect class of injectivity with its generalizations. Second: Study the topic of this article contributes to the improvement of the vision for finding the corresponding between acts theory and module theory. Third: This article has dealt with the important subject in the field of science and knowledge especially in algebra and can take it as a basis for future work for the researchers who work on algebra. Now, in this paper, the concept of closed quasi injective acts over monoids is introduced which represents a generalization of quasi injective. Several characterizations of this concept are given to show the behavior of the property of closed quasi injective. Relationship of the concept of closed quasi injective acts over monoids with Hopfian, co-Hopfian and directly finite property are considered. This work gives the answer to the question of what are the conditions to be met in the subacts in order to inherit the property of closed quasi injectivity. We obtained the main result in this direction in proposition (2.5) and proposition (2.6). A part of this paper was devoted to studying the relationship among the class of closed quasi injective acts with some generalizations of injectivity.

 Published in Pure and Applied Mathematics Journal (Volume 8, Issue 5) This article belongs to the Special Issue Algebra with Its Applications DOI 10.11648/j.pamj.20190805.12 Page(s) 88-92 Creative Commons This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. Copyright Copyright © The Author(s), 2019. Published by Science Publishing Group
Keywords

Closed Quasi Injective Acts, Extending Acts, Continuous Acts, Noetherian Acts, Hopfian Acts

References
• APA Style

Shaymaa Amer Abdul-Kareem, Ahmed Amer Abdulkareem. (2019). About the Closed Quasi Injective S-Acts Over Monoids. Pure and Applied Mathematics Journal, 8(5), 88-92. https://doi.org/10.11648/j.pamj.20190805.12

ACS Style

Shaymaa Amer Abdul-Kareem; Ahmed Amer Abdulkareem. About the Closed Quasi Injective S-Acts Over Monoids. Pure Appl. Math. J. 2019, 8(5), 88-92. doi: 10.11648/j.pamj.20190805.12

AMA Style

Shaymaa Amer Abdul-Kareem, Ahmed Amer Abdulkareem. About the Closed Quasi Injective S-Acts Over Monoids. Pure Appl Math J. 2019;8(5):88-92. doi: 10.11648/j.pamj.20190805.12

• ```@article{10.11648/j.pamj.20190805.12,
author = {Shaymaa Amer Abdul-Kareem and Ahmed Amer Abdulkareem},
title = {About the Closed Quasi Injective S-Acts Over Monoids},
journal = {Pure and Applied Mathematics Journal},
volume = {8},
number = {5},
pages = {88-92},
doi = {10.11648/j.pamj.20190805.12},
url = {https://doi.org/10.11648/j.pamj.20190805.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20190805.12},
abstract = {The aim of introducing and studying the notion of closed quasi injective S-act is to create a basis facilitate for the exchange ideas between module theory and act theory. As well as it represents a generalization of the quasi-injective act. The quasi-injective act was first introduced and studied by A. M. Lopez, Jr. and J. K. Luedeman, 1979. Then the author was one of the researchers which introduced several generalizations for this notion from several aspects because of its importance. More accurately, the contribution of this paper to the field of competence can be summarized into three points as follows: First: The possibilities for applying the topic of this article helps researchers about how can connect class of injectivity with its generalizations. Second: Study the topic of this article contributes to the improvement of the vision for finding the corresponding between acts theory and module theory. Third: This article has dealt with the important subject in the field of science and knowledge especially in algebra and can take it as a basis for future work for the researchers who work on algebra. Now, in this paper, the concept of closed quasi injective acts over monoids is introduced which represents a generalization of quasi injective. Several characterizations of this concept are given to show the behavior of the property of closed quasi injective. Relationship of the concept of closed quasi injective acts over monoids with Hopfian, co-Hopfian and directly finite property are considered. This work gives the answer to the question of what are the conditions to be met in the subacts in order to inherit the property of closed quasi injectivity. We obtained the main result in this direction in proposition (2.5) and proposition (2.6). A part of this paper was devoted to studying the relationship among the class of closed quasi injective acts with some generalizations of injectivity.},
year = {2019}
}
```
• ```TY  - JOUR
T1  - About the Closed Quasi Injective S-Acts Over Monoids
AU  - Shaymaa Amer Abdul-Kareem
AU  - Ahmed Amer Abdulkareem
Y1  - 2019/10/24
PY  - 2019
N1  - https://doi.org/10.11648/j.pamj.20190805.12
DO  - 10.11648/j.pamj.20190805.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 88
EP  - 92
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20190805.12
AB  - The aim of introducing and studying the notion of closed quasi injective S-act is to create a basis facilitate for the exchange ideas between module theory and act theory. As well as it represents a generalization of the quasi-injective act. The quasi-injective act was first introduced and studied by A. M. Lopez, Jr. and J. K. Luedeman, 1979. Then the author was one of the researchers which introduced several generalizations for this notion from several aspects because of its importance. More accurately, the contribution of this paper to the field of competence can be summarized into three points as follows: First: The possibilities for applying the topic of this article helps researchers about how can connect class of injectivity with its generalizations. Second: Study the topic of this article contributes to the improvement of the vision for finding the corresponding between acts theory and module theory. Third: This article has dealt with the important subject in the field of science and knowledge especially in algebra and can take it as a basis for future work for the researchers who work on algebra. Now, in this paper, the concept of closed quasi injective acts over monoids is introduced which represents a generalization of quasi injective. Several characterizations of this concept are given to show the behavior of the property of closed quasi injective. Relationship of the concept of closed quasi injective acts over monoids with Hopfian, co-Hopfian and directly finite property are considered. This work gives the answer to the question of what are the conditions to be met in the subacts in order to inherit the property of closed quasi injectivity. We obtained the main result in this direction in proposition (2.5) and proposition (2.6). A part of this paper was devoted to studying the relationship among the class of closed quasi injective acts with some generalizations of injectivity.
VL  - 8
IS  - 5
ER  - ```
Author Information
• Department of Mathematics, College of Basic Education, Mustansiriyah University, Baghdad, Iraq

• Department of Computer Science, College of Science, Mustansiriyah University, Baghdad, Iraq

• Sections