On sum annihilator ideals in Ore extensions

Year 2024, Volume: 53 Issue: 3, 704 - 713, 27.06.2024

Mahsa Paykanian , Ebrahim Hashemi , Abdollah Alhevaz

https://doi.org/10.15672/hujms.1037521

Abstract

A ring $R$ is called a left Ikeda-Nakayama ring (left IN-ring) if the right annihilator of the intersection of any two left ideals is the sum of the two right annihilators. As a generalization of left IN-rings, a ring $R$ is called a right SA-ring if the sum of right annihilators of two ideals is a right annihilator of an ideal of $R$. It would be interesting to find conditions under which an Ore extension $R[x; \alpha, \delta]$ is IN and SA. In this paper, we will present some necessary and sufficient conditions for the Ore extension $R[x;\alpha, \delta]$ to be left IN or right SA. In addition, for an $(\alpha,\delta)$-compatible ring $R$, it is shown that: (i) If $S = R[x;\alpha,\delta]$ is a left IN-ring with ${\rm{Idm}}(R) ={\rm{Idm}}(R[x;\alpha, \delta])$, then $R$ is left McCoy. (ii) Every reduced left IN-ring with finitely many minimal prime ideals is a semiprime left Goldie ring. (iii) If $R$ is a commutative principal ideal ring, then $R$ and $R[x]$ are IN. (iv) If $R$ is a reduced ring and $n$ is a positive integer, then $R$ is right SA if and only if $R[x]/(x^{n+1})$ is right SA.

Keywords

Armendariz ring, Ikeda-Nakayama ring, quasi-Armendariz ring, SA-ring, skew polynomials ring, left McCoy ring

References

• [1] D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26, 2265-2272, 1998.
• [2] E. P. Armendariz, H. K. Koo and J. K. Park, Isomorphic Ore extensions, Comm. Algebra 15, 2633-2652, 1987.
• [3] G. F. Birkenmeier, M. Ghirati and A. Taherifar, When is a sum of annihilator ideals an annihilator ideal?, Comm. Algebra 43, 2690-2702, 2015.
• [4] G. F. Birkenmeier, M. Ghirati, A. Ghorbani, A. Naghdi and A. Taherifar, Corrigendum to: When is a sum of annihilator ideals an annihilator ideal?, Comm. Algebra 46 (10), 4174-4175, 2018.
• [5] V. Camillo, W. K. Nicholson and M. F. Yousif, Ikeda-Nakayama rings, J. Algebra 226, 1001-1010, 2000.
• [6] K. R. Goodearl and R. B. Warfield Jr, An Introduction to Noncommutative Noetherian Rings, London Mathematical Society Student Texts 61, 2nd ed. Cambridge: Cambridge University Press, 2004.
• [7] C. R. Hajarnavis and N. C. Norton, On dual rings and their modules, J. Algebra 93, 253-266, 1985.
• [8] E. Hashemi, Compatible ideals and radicals of Ore extensions, New York J. Math. 12, 349-356, 2006.
• [9] E. Hashemi and A. Moussavi, Polynomial extensions of quasi-Baer rings, Acta Math. Hungar. 107, 207-224, 2005.
• [10] E. Hashemi, M. Hamidizadeh and A. Alhevaz, Some types of ring elements in Ore extensions over noncommutative rings, J. Algebra Appl. 16 (11), 1750201, 2017.
• [11] Y. Hirano, On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra 168, 45-52, 2002.
• [12] A. A. M. Kamal, Idempotents in polynomial rings, Acta Math. Hungar. 59 (3-4), 355-363, 1992.
• [13] I. Kaplansky, Dual rings, Ann. of Math. 49, 689-701, 1948.
• [14] A. Leroy and J. Matczuk, Goldie conditions for ore extensions over semiprime rings, Algebr. Represent. Theory 8 (5), 679-688, 2005.
• [15] J. C. McConnell, J. C. Robson and L. W. Small, Noncommutative Noetherian Rings, Vol. 30. Providence, Rhode Island: American Mathematical Society, 2001.
• [16] N. H. McCoy, Remarks on divisors of zero, Amer. Math. Monthly 49, 286-295, 1942.
• [17] P. P. Nielsen, Semi-commutativity and the McCoy condition, J. Algebra 298, 134-141, 2006.
• [18] M. Paykanian, E. Hashemi and A. Alhevaz, On skew polynomials over Ikeda- Nakayama rings, Comm. Algebra 49 (9), 4038-4049, 2021.
• [19] R. Wisbauer, M. F. Yousif and Y. Zhou, Ikeda-Nakayama modules, Beitr. Algebra Geom. 43, 111-119, 2002.
• [20] O. Zariski and P. Samuel, Commutative Algebra, volume I, Van Nostrand, Princeton, 1960.