Hacettepe Journal of Mathematics and Statistics 2651-477X 2651-477X Hacettepe University 10.15672/hujms.592974 Mathematical Sciences Matematik Quasi-$n$-absorbing and semi-$n$-absorbing preradicals https://orcid.org/0000-0002-1690-0607 Mostafanasab Hojjat https://orcid.org/0000-0002-7411-8621 Yousefian Darani Ahmad 10 08 2019 48 5 1286 1303 03 16 2015 02 20 2018 Copyright © 2002, Hacettepe Journal of Mathematics and Statistics 2002 Hacettepe Journal of Mathematics and Statistics

The aim of this paper is to introduce the notions of quasi-$n$-absorbing preradicals and of semi-$n$-absorbing preradicals. These notions are inspired by applying the concept of $n$-absorbing preradicals to semiprime preradicals. Also, we study the concepts of quasi-$n$-absorbing submodules and of semi-$n$-absorbing submodules and their relations with quasi-$n$-absorbing preradicals and semi-$n$-absorbing preradicals.

 lattice preradical quasi-$n$-absorbing semi-$n$-absorbing [1] D.F. Anderson and A. Badawi, On n-absorbing ideals of commutative rings, Comm. Algebra 39, 1646–1672, 2011. [2] A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc. 75, 417–429, 2007. [3] J. Beachy, M-injective modules and prime M-ideals, Comm. Algebra 30 (10), 4649– 4676, 2002. [4] L. Bican, P. Jambor, T. Kepka and P. Nemec, Preradicals, Comment. Math. Univ. Carolinae 15 (1), 75–83, 1974. [5] L. Bican, P. Jambor, T. Kepka and P. Nemec, Prime and coprime modules, Fund. Math. 107, 33–45, 1980. [6] L. Bican, T. Kepka, and P. Nemec, Rings, Modules and Preradicals, Marcel Dekker, New York, 1982. [7] A.I. Kashu, On some operations in the lattice of submodules determined by preradi- cals, Bull. Acad. Stiinte Repub. Mold. Mat. 2 (66), 5–16, 2011. [8] T.Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, Springer-Verlag, 1998. [9] F. Raggi, J. Ríos, H. Rincón and R. Fernández-Alonso, Basic preradicals and main injective modules, J. Algebra Appl. 8 (1), 1–16, 2009. [10] F. Raggi, J. Ríos, H. Rincón, R. Fernández-Alonso and C. Signoret, The lattice structure of preradicals, Comm. Algebra 30 (3), 1533–1544, 2002. [11] F. Raggi, J. Ríos, H. Rincón, R. Fernández-Alonso and C. Signoret, The lattice structure of preradicals II : partitions, J. Algebra Appl. 1 (2), 201–214, 2002. [12] F. Raggi, J. Ríos, H. Rincón, R. Fernández-Alonso and C. Signoret, The lattice structure of preradicals III : operators, J. Pure and Applied Algebra 190, 251–265, 2004. [13] F. Raggi, J. Ríos, H. Rincón, R. Fernández-Alonso and C. Signoret, Prime and irreducible preradicals, J. Algebra Appl. 4 (4), 451–466, 2005. [14] F. Raggi, J. Ríos, H. Rincón, R. Fernández-Alonso and C. Signoret, Semiprime preradicals, Comm. Algebra 37, 2811–2822, 2009. [15] B. Stenström, Rings of Quotients, Die Grundlehren der Mathematischen Wis- senschaften, Band 217, Springer Verlag, Berlin, 1975. [16] D. Keskin Tütüncü and Y. Kuratomi, On mono-injective modules and mono-ojective modules, Math. J. Okayama Univ. 55, 117–129, 2013. [17] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Philadelphia, 1991. [18] A. Yousefian Darani, and H. Mostafanasab, Co-2-absorbing preradicals and submod- ules, J. Algebra Appl. 14 (7), 1550113 (23 pages), 2015. [19] A. Yousefian Darani and H. Mostafanasab, On 2-absorbing preradicals, J. Algebra Appl. 14 (2), 1550017 (22 pages), 2015. [20] A. Yousefian Darani and F. Soheilnia, 2-absorbing and weakly 2-absorbing submod- uels, Thai J. Math. 9 (3), 577–584, 2011.