Max CS and min CS modules

In this work, we study max CS, min CS, max-min CS modules and their endomorphism rings. Under certain conditions e.g., related to nonsingularity and duo-ness , we prove that a module is max CS if and only if it is min CS, and that direct sums of min max CS modules is again min max CS. Finally, symmetry of max-min CS property on the endomorphism rings of max-min CS modules is investigated.

Keywords:

Max CS modules, min CS modules, max-min CS rings nonsingular modules, duo modules,

___

[1] Goodearl KR. Ring Theory: Nonsingular Rings and Modules. Boca Raton, Florida, USA: CRC Press, 1976.
[2] Jain SK, Husain SA, Adel NA. Right-Left Symmetry of Right Nonsingular Right Max-Min CS Prime Rings. Communications in Algebra 2006; 34: 3883-3889. doi: 10.1080/00927870600862714
[3] Hadi IMA, Majeed RN. Min (max)-CS modules. Ibn Al-Haitham Journal for Pure and Applied Science 2012; 25 (1).
[4] Khuri SM. Endomorphism rings of nonsingular modules. Annales mathématiques du Québec 1980; 4: 145-152.
[5] Khuri SM. Modules whose endomorphism rings have isomorphic maximal left and right quotient rings. Proceedings of the American Mathematical Society 1982; 85 (2): 161-164. doi: 10.1090/S0002-9939-1982-0652433-5
[6] Khuri SM. Nonsingular retractable modules and their endomorphism rings. Bulletin of the Australian Mathematical Society 1991; 43: 63-71. doi: 10.1017/S000497270002877X
[7] Sanh NV, Vu NA, Asawasamrit S, Ahmed KFU, Thao LP. Primeness in module category. Asian-European Journal of Mathematics 2010; 3 (1): 145-154. doi: 10.1142/S1793557110000106
[8] Thuat DV, Hai HD, Nghiem NDH, Sarapee C. On the endomorphism rings of Max CS and Min CS Modules. AIP Conference Proceedings 2016; 1775: 030066. doi: 10.1063/1.4965186
[9] Thuat D, Hai DH, Tu TD. Symmetry of extending properties in nonsingular Utumi rings. Surveys in Mathematics and its Application 2020; 15: 281-293.
[10] Tercan A, Yücel CC. Module Theory, Extending Modules and Generalizations. Basel, Switzerland: Birkhäuser, 2016.