On the generalization of torsion functor and P-semiprime modules over noncommutative rings

Document Type : Research Paper

Authors

1 Department of Mathematics, Addis Ababa University

2 The Namibia University of Science and Technology, Namibia

Abstract

Let R be an associative Noetherian unital noncommutative ring R. We introduce the functor PΓP over the category of R-modules and use it to characterize P-semiprime. P-semisecond R-modules also characterized by the functor PΛP. We also show that the Greenless-May Duality (GM) and Matlis Greenless-May Equality(MGM) hold over the full subcategory of R-Mod consisting of R-semiprime and R-semisecond modules. Finally, we generate a one-sided right ideal PΓP(R), which gives an equivalent formulation to solve K{\"o}the conjecture positively or negatively.

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Main Subjects


[1] S. Agata, On some results related to K{\"o}the's conjecture, Serdica Math. J., {\bf 27} (2) (2001), 159--170.
[2] R. Beyranvand and F. Rastgoo, Weakly second modules over noncommutative rings, Hacettepe J. Math. Stat., {\bf 45} (5) (2016), 1355--1366.
[3] M. P. Brodmann and R. Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge univ. press {\bf 136} (2012).
[4] T. Cheatham and E. Enochs, Injective hulls of flat modules, Comm. Algebra, {\bf 8}(20) (1980), 1989--1995.
[5] D. J. Fieldhouse, Pure theories, Math. Ann. {\bf 184} (1969), 1–18.
[6] J. Garc{\'\i}a and J. M. Hern{\'a}ndez, When is the category of flat modules abelian?, Fundam. Math, {\bf 147} (1) (1995), 83--91.
[7] S. J{\o}ndrup and D. Simson, Indecomposable modules over semiperfect rings, J. Algebra, {\bf 73} (1) (1981), 23--29.
[8] G. K{\"o}ethe. Die Struktur der Ringe, deren Restklassenring nach dem Radikal vollstanding irreduzibel ist. Math. Z. {\bf 32} (1930), 161–186.
[9] A. Kyomuhangi and D. Ssevviiri, The locally nilradical for modules over commutative rings, Beitr. Algebra Geom. {\bf 61}(4) (2020), 759--769.
[10] T. Y. Lam, A first course in noncommutative rings, {\bf 131} (1991), Springer.
[11] T. K. Lee and Y. Zhou, Reduced modules, rings, modules, algebras and abelian groups, Lecture Notes in Pure and Appl. Math. {\bf 236} (2004), 365--377.
[12] L. N{\v{e}}mec, T. Bican, P. Kepka, Rings, modules and preradicals, Lect. notes in pure and appl. math. {\bf 75} (1982).
[13] M. Porta, L. Shaul and A. Yekutieli, On the homology of completion and torsion, Algebr. Represent. Theory, 17(1) (2014), 31-67.
[14] E. R. Puczy{\l}owski, Questions related to K{\"o}ethe's nil ideal problem, Algebra and its applications, contemporary mathematics, {\bf 419} (2006), 269--283.
[15] S. P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Commun, Algebra, {\bf 31}(9) (2003), 4425--4443.
[16] M. B. Rege and A. M. Buhphang, On reduced modules and rings, Int. Elect. J. Algebra {\bf 3} (2008), 58--74.
[17] J. J. Rotman, An introduction to homological algebra, {\bf 2} (2009), Springer.
[18] D. Ssevviiri Applications of reduced and coreduced modules II, arXiv preprint arXiv:2205.13241, (2023).
[19] D. Ssevviiri, Applications of reduced and coreduced modules I, Int. Electron. J. Algebra (2023), DOI: 10.24330/ieja.1299587.
[20] B. Stenstr{\"o}m, Rings of quotients: an introduction to methods of ring theory, {\bf 217} (2012), Springer Science \& Business Media.
[21] H. Tachikawa, QF-3 rings and categories of projective modules, J. Algebra, {\bf 28} (1974), 408--413.
[22] D. Ssevviiri and N. Groenewald, Generalization of nilpotency of ring elements to module elements, Commun. Algebra, {\bf 42} (2) (2014), 571--577.