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Volume 8, Issue 5, October 2019, Page: 88-92
About the Closed Quasi Injective S-Acts Over Monoids
Shaymaa Amer Abdul-Kareem, Department of Mathematics, College of Basic Education, Mustansiriyah University, Baghdad, Iraq
Ahmed Amer Abdulkareem, Department of Computer Science, College of Science, Mustansiriyah University, Baghdad, Iraq
Received: Sep. 8, 2019;       Accepted: Oct. 15, 2019;       Published: Oct. 24, 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.
Keywords
Closed Quasi Injective Acts, Extending Acts, Continuous Acts, Noetherian Acts, Hopfian Acts
Shaymaa Amer Abdul-Kareem, Ahmed Amer Abdulkareem, About the Closed Quasi Injective S-Acts Over Monoids, Pure and Applied Mathematics Journal. Special Issue: Algebra with Its Applications. Vol. 8, No. 5, 2019, pp. 88-92. doi: 10.11648/j.pamj.20190805.12
Reference
[1]
M. Kilp, U. Knauer, & A. V. Mikhalev, “Monoids acts and categories with applications to wreath products and graphs”, a handbook for students and researchers, Berlin, W. de Gruyter, (2000). Retrieved from https://www.amazon.com/Categories-Applications-Expositions-Mathematics-Degruyter/dp/3110152487
[2]
[3]
A. Shaymaa, “Generalizations of quasi injective acts over monoids”, PhD. thesis, College of Science, University of Al-Mustansiriyah, Baghdad, Iraq, (2015). Retrieved from https://www.academia.edu/35100206/GENERALIZATIONS_OF_QUASI_INJECTIVE_S-SYSTEMS_OVER_MONOIDS
[4]
A. Shaymaa, “Pseudo C-M-injective and pseudo C-quasi principally injective acts over monoids”, Journal of Progressive Research in Mathematics (JPRM), 6 (3), pp. 788-802, (2016). Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/580
[5]
A. Shaymaa, “Pseudo IC-injective systems over monoids”, Global Journal of mathematics (GJM), 6 (3), pp. 636-642, (2016). Retrieved from https://www.researchgate.net/publication/324835936_PSEUDO_IC-INJECTIVE_SYSTEMS_OVER_MONOIDS
[6]
Berthiaume P., “The injective envelope of S-sets”, Canad. Math. Bull., 10 (2), pp. 261–272, (1967). Retrieved from https://www.cambridge.org/core/services/aop-cambridge-core/content/view/CC5C688B45202403E48AE8B5C75CEC0F/S0008439500067242a.pdf/injective_envelope_of_ssets.pdf
[7]
Shaymaa A., “On Finitely Generated in S-systems over monoids”, Noor Publishing, Germany, (2018). Retrieved from https://www.amazon.com/s?k=On+Finitely+Generated+in+S-systems+over+Monoids&ref=nb_sb_noss
[8]
T.Yan, “Generalized injective S-acts on a monoid”, Advances in mathematics, 40 (4), pp. 421-432, (2011). Retrieved from http://sci.upc.edu.cn/2016/0926/c6900a96616/page.htm
[9]
K. Sungjin and K. Jupil, “Weakly large subacts of S-act”, J. of Chungcheong Math. Soc., 20 (4), pp486-493, (2007).
[10]
C. V. Hinkle and Jr., “The extended centralizer of an S-set”, Pacific journal of mathematics, 53 (1), pp163-170, (1974). Retrieved from https://msp.org/pjm/1974/53-1/pjm-v53-n1-p14-s.pdf
[11]
J. Ahsan, “Monoids characterized by their quasi injective S-acts”, Semigroup forum, 36 (3), pp 285-292, (1987). Retrieved from https://link.springer.com/article/10.1007%2FBF02575022
[12]
A. M. Lopez, Jr. and J. K. Luedeman, “Quasi-injective S-acts and their S-endomorphism Semigroup”, Czechoslovak Math. J., 29 (104), pp97-104, (1979). Retrieved from https://dml.cz/bitstream/handle/10338.dmlcz/101581/CzechMathJ_29-1979-1_11.pdf
[13]
M. S. Abbas and A. Shaymaa, “Principally quasi injective act over monoid”, Journal of advances in mathematics, 10 (1), 3152-3162, (2015). Retrieved from https://www.researchgate.net/publication/324845514_PRINCIPALLY_QUASI_INJECTIVE_SYSTEM_OVER_MONOID
[14]
A. Shaymaa, “Extending and P-extending S-act over Monoids”, International Journal of Advanced Scientific and Technical Research, 2 (7), pp. 171-178, (2017). Retrieved from https://www.academia.edu/34905983/Extending_and_P-extending_S-act_over_Monoids
[15]
A. K. Tiwary, S. A. Paramhans, and B. M. Pandeya, “Generalization of quasi injectivity”, Progress of Mathematics, 13 (1-2), pp. 31-40, (1979).