SOME PROPERTIES OF PRIME MODULES AND SEMIPRIME MODULES
DOI:
https://doi.org/10.35747/t.v5i1.573Keywords:
prime modules, semiprime modules, fully prime (semiprime) modulesAbstract
Prime modules was introduced by Feller and Swokowski. In J. Dauns did a research about generalization of prime modules and defined semiprime modules as weaker terms of prime modules. In other hand, Abu-Saymeh studied characteristic ring of prime modules. Furthermore Behboodi, Karamzadeh, and Koohy defined fully prime (semiprime) modules. Necessary and sufficient conditions for those type of modules was also given. In this articles, we will discuss the result of Behboodi, et al. those are necessary and sufficient condition for fully prime (semiprime) modules. Then the relation between prime modules and semiprime modules is studied. At the end, we will prove a new theorem about necessary and sufficient conditions for semiprime modules to be a prime modules with added some condition to semiprime modules’s annihilator.
References
Abu-Saymeh, S. (1995). On Dimensions of Finitely Generated Modules. Communications in Algebra. https://doi.org/10.1080/00927879508825270
Behboodi, M., Karamzadeh, O. A. S., & Koohy, H. (2004). Modules whose certain submodules are prime. Vietnam Journal of Mathematics, 32(3), 303–317.
Behboodi, M., Shojaee, S.H. (2012): Commutative local rings whose ideals are direct sums of cyclics
Dauns, J. (1978). Prime modules. Journal Fur Die Reine Und Angewandte Mathematik. https://doi.org/10.1515/crll.1978.298.156
Dummit, D. S. (2004): Abstract Algebra (3rd ed.), John Wiley and Sons.
Feller, E. H., & Swokowski, E. W. (1965). Prime Modules. Canadian Journal of Mathematics. https://doi.org/10.4153/cjm-1965-099-5
F. Raggi, J. Rios, H. Rincon, R. Fernandez-Alonso, and C. Signoret, Prime and irreducible preradicals, J. Algebra Appl. 4 (2005), no. 4, 451-466
F. Raggi, J. Rios, H. Rincon, R. Fernandez-Alonso, and C. Signoret, Semiprime preradicals, Comm. Algebra 37 (2009), no. 8, 2811-2822.
Jacobson, N. (1943): The Teory of Rings, Mathematical Surveys and Monographs, American Mathematical Society.
Khashan, H.A. (2012): On almost prime submodules, Acta Mathematica Scientia, 32(2); 645-651
Lam, T. Y. (1991): A First Course in Noncommutative Rings, Graduate Texts in Mathematics, 131. Springer-Verlag, 31-37
M. Medina-Barcenas and A. C. Ozcan, Primitive submodules, co-semisimple and regular modules, Taiwanese J. Math. 22 (2018), no. 3, 545-565
R. Wisbauer, Foundations of module and ring theory, revised and translated from the 1988 German edition, Algebra, Logic and Applications, 3, Gordon and Breach Science Publishers, Philadelphia, PA, 1991
R. Wisbauer, Modules and algebras: bimodule structure and group actions on algebras, Pitman Monographs and Surveys in Pure and Applied Mathematics, 81, Longman, Harlow, 1996.
Tavallaee, H.A., Varmazyar, R. (2008): Semi-radicals of submodules in modules, IUST International Journal of Engineering Science, 19(1); 21-27
