Generalizations of 2-absorbing and 2-absorbing primary submodules

Year 2019, Volume: 48 Issue: 4, 1001 - 1016, 08.08.2019

Hojjat Mostafanasab Ünsal Tekir Ece Yetkin Çelikel Emel A. Uğurlu Gülşen Ulucak Ahmad Yousifean Darani

Abstract

In this study, we introduce $\phi $-2-absorbing and $\phi $-2-absorbing primary submodules of modules over commutative rings generalizing the concepts of 2-absorbing and 2-absorbing primary submodules. Let $\phi :S(M)\rightarrow S(M)\cup \{\emptyset \}$ be a function where $S(M)$ denotes the set of all submodules of $M$ and $N$ a proper submodule of an $R$-module $M$. We will say that $N$ is a $\phi $-\textit{2-absorbing submodule} of $M$ if whenever $a,b\in R$, $m\in M$ with $abm\in N$ and $abm\notin \phi (N)$, then $am\in N$ or $bm\in N$ or $ab\in (N:_{R}M)$ and $N$ is said to be a $\phi $-2-absorbing primary submodule of $M$ whenever if $a,b\in R$, $m\in M$ with $abm\in N$ and $abm\notin \phi (N)$, then $am\in M$-$\mathrm{rad}(N)$ or $bm\in M$-$\mathrm{rad}(N)$ or $ab\in (N:_{R}M)$. We investigate many properties of these new types of submodules and establish some characterizations for $\phi $-2-absorbing and $\phi $-2-absorbing primary submodules of multiplication modules.

Keywords

$\phi $-prime submodule, $\phi $-primary submodule, 2-absorbing primary submodule, weakly 2-absorbing primary submodule, $\phi $-2-absorbing submodule, $\phi $-2-absorbing primary submodule

References

• [1] M.M. Ali, Idempotent and nilpotent submodules of multiplication modules, Comm. Algebra 36, 4620-4642, 2008.
• [2] R. Ameri, On the prime submodules of multiplication modules, Int. J. Math. Math. Sci. 27, 1715-1724, 2003.
• [3] D.F. Anderson and A. Badawi, On n-absorbing ideals of commutative rings, Comm. Algebra 39, 1646-1672, 2011.
• [4] D.D. Anderson and M. Batanieh, Generalizations of prime ideals, Comm. Algebra 36, 686-696, 2008.
• [5] D.D. Anderson and E. Smith, Weakly prime ideals, Houston J. Math. 29, 831-840, 2003.
• [6] H. Ansari-Toroghy and F. Farshadifar, Fully idempotent and coidempotent submodules, Bull. Iranian Math. Soc. 38 (4), 987-1005, 2012.
• [7] S.E. Atani, F. Callialp and U. Tekir, A Short note on the primary submodules of multiplication modules, Int. J. Algebra 8 (1), 381-384, 2007.
• [8] S.E. Atani and F. Farzalipour, On weakly prime submodules, Tamk. J. Math. 38 (3), 247-252, 2007.
• [9] A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc. 75, 417-429, 2007.
• [10] A. Badawi and A.Y. Darani, On weakly 2-absorbing ideals of commutative rings, Houston J. Math. 39, 441-452, 2013.
• [11] A. Badawi, U. Tekir, E.A. Ugurlu, G. Ulucak and E. Yetkin Celikel, Generalizations of 2-absorbing primary ideals of commutative rings, Turk. J. Math. 40, 703-717, 2016.
• [12] A. Badawi, U. Tekir and E. Yetkin, On 2-absorbing primary ideals in commutative rings, Bull. Korean Math. Soc. 51 (4), 1163-1173, 2014
• [13] A. Barnard, Multiplication modules, J. Algebra, 71, 174-178, 1981.
• [14] M. Batanieh and K. Dofa, Generalizations of primary ideals and submodules, Int. J. Contemp. Math. Sciences 6, 811-824, 2011.
• [15] M. Behboodi and H. Koohi, Weakly prime modules, Vietnam J. Math. 32 (2), 185-195, 2004.
• [16] A.Y. Darani, Generalizations of primary ideals in commutative rings, Novi Sad J. Math. 42, 27-35, 2012.
• [17] A.Y. Darani and F. Soheilnia, On 2-absorbing and weakly 2-absorbing submodules, Thai J. Math. 9, 577-584, 2011.
• [18] A.Y. Darani and F. Soheilnia, On n-absorbing submodules, Math. Comm. 17, 547-557, 2012.
• [19] M. Ebrahimpour and R. Nekooei, On generalizations of prime ideals, Comm. Algebra 40 1268-1279, 2012.
• [20] Z.A. El-Bast and P.F. Smith, Multiplication modules, Comm. Algebra 16, 755-779, 1988.
• [21] A. Khaksari and A. Jafari, ϕ-prime submodules, Int. J. Algebra 5 (29), 1443-1449, 2011.
• [22] R.L. McCasland and M.E. Moore, Radicals of submodules, Comm. Algebra 19, 1327- 1341, 1991.
• [23] H. Mostafanasab, E. Yetkin, U. Tekir and A.Y. Darani, On 2-absorbing primary submodules of modules over commutative rings, An. Şt. Univ. "Ovidius" Constanta Ser. Mat. 24 (1), 335-351, 2016.
• [24] P. Quartararo and H.S. Butts, Finite unions of ideals and modules, Proc. Amer. Math. Soc. 52, 91-96,1975.
• [25] P.F. Smith, Some remarks on multiplication modules, Arch. Math. 50, 223-235, 1988.
• [26] N. Zamani, ϕ-prime submodules, Glasgow Math. J., 52 (2), 253-259, 2010.