Mathematica Bohemica

Mathematica Bohemica, first online, pp. 1-9

Some annihilator ideals in skew Hurwitz series rings

Amit B. Singh, Deepa Arora

Received June 13, 2024. Published online January 17, 2025.

Abstract: A ring $R$ has right (left) property (A) if for every finitely generated two-sided ideal $I\subseteq Z_l(R)$ $(I\subseteq Z_r(R))$, there exists nonzero $u\in R$ $(v\in R)$ such that $Iu=0$ $(vI=0)$. In this article, we establish a relationship between a ring with property (A) and its skew Hurwitz series ring $(H\!R, \omega)$, where $\omega$ is an endomorphism of $R$. Also some properties of strongly right AB ring for skew Hurwitz series rings are studied.

Keywords: ring with right property (A); skew Hurwitz series ring; $\omega$-compatible ring

Classification MSC: 16D25, 16D70, 16S34

DOI: 10.21136/MB.2025.0066-24

PDF available at: Institute of Mathematics CAS

References:
[1] M. Ahmadi, A. Moussavi, V. Nourozi: On skew Hurwitz serieswise Armendariz rings. Asian-Eur. J. Math. 7 (2014), Article ID 1450036, 19 pages. DOI 10.1142/S1793557114500363 | MR 3257511 | Zbl 1308.16033
[2] D. D. Anderson, V. Camillo: Armendariz rings and Gaussian rings. Commun. Algebra 26 (1998), 2265-2272. DOI 10.1080/00927879808826274 | MR 1626606 | Zbl 0915.13001
[3] S. Annin: Associated primes over Ore extension rings. J. Algera Appl. 3 (2004), 193-205. DOI 10.1142/S0219498804000782 | MR 2069261 | Zbl 1060.16029
[4] F. Azarpanah, O. A. S. Karamzadeh, A. R. Aliabad: On ideals consisting entirely of zero divisors. Commun. Algebra 28 (2000), 1061-1073. DOI 10.1080/00927870008826878 | MR 1736781 | Zbl 0970.13002
[5] A. Benhissi, F. Koja: Basic properties of Hurwitz series rings. Ric. Mat. 61 (2012), 255-271. DOI 10.1007/s11587-012-0128-2 | MR 3000659 | Zbl 1318.13034
[6] S. Bochner, W. T. Martin: Singularities of composite functions in several variables. Ann. Math. (2) 38 (1937), 293-302. DOI 10.2307/1968554 | MR 1503336 | Zbl 0016.31103
[7] C. Faith: Algebra II. Ring Theory. Grundlehren der mathematischen Wissenschaften 191. Springer, Berlin (1976). DOI 10.1007/978-3-642-65321-6 | MR 0427349 | Zbl 0335.16002
[8] M. Fliess: Sur divers produits de séries formelles. Bull. Soc. Math. Fr. 102 (1974), 181-191. (In French.) DOI 10.24033/bsmf.1777 | MR 0354647 | Zbl 0313.13021
[9] E. Hashemi, A. A. Estaji, A. Alhevaz: On Ore extension and skew power series rings with some restrictions on zero-divisors. J. Algebra Appl. 16 (2017), Article ID 1750164, 12 pages. DOI 10.1142/S021949881750164X | MR 3661631 | Zbl 1382.16022
[10] E. Hashemi, A. A. Estaji, M. Ziembowski: Answers to some questions concerning rings with property $(A)$. Proc. Edinb. Math. Soc., II. Ser. 60 (2017), 651-664. DOI 10.1017/S0013091516000407 | MR 3674083 | Zbl 1405.16036
[11] E. Hashemi, A. Moussavi: Polynomial extensions of quasi-Baer rings. Acta. Math. Hung. 107 (2005), 207-224. DOI 10.1007/s10474-005-0191-1 | MR 2148584 | Zbl 1081.16032
[12] A. M. Hassanein: Clean rings of skew Hurwitz series. Matematiche 62 (2007), 47-54. MR 2389111 | Zbl 1150.16029
[13] A. M. Hassanein, M. A. Farahat: Some properties of skew Hurwitz series. Matematiche 69 (2014), 169-178. MR 3215878 | Zbl 1320.16023
[14] M. Henriksen, M. Jerison: The space of minimal prime ideals of a commutative ring. Trans. Am. Math. Soc. 115 (1965), 110-130. DOI 10.1090/S0002-9947-1965-0194880-9 | MR 0194880 | Zbl 0147.29105
[15] G. W. Hinkle, J. A. Huckaba: The generalized Kronecker function ring and the ring $R(X)$. J. Reine Angew. Math. 292 (1977), 25-36. DOI 10.1515/crll.1977.292.25 | MR 0447213 | Zbl 0348.13011
[16] C. Y. Hong, N. K. Kim, T. K. Kwak: On skew Armendariz rings. Commun. Algebra 31 (2003), 103-122. DOI 10.1081/AGB-120016752 | MR 1969216 | Zbl 1042.16014
[17] C. Y. Hong, N. K. Kim, Y. Lee, S. J. Ryu: Rings with property $(A)$ and their extensions. J. Algebra 315 (2007), 612-628. DOI 10.1016/j.jalgebra.2007.01.042 | MR 2351882 | Zbl 1156.16001
[18] J. A. Huckaba, J. M. Keller: Annihilation of ideals in commutative rings. Pac. J. Math. 83 (1979), 375-379. DOI 10.2140/pjm.1979.83.375 | MR 0557938 | Zbl 0388.13001
[19] A. Hurwitz: Sur un théorème de M. Hadamard. C. R. Acad. Sci., Paris 128 (1899), 350-353. (In French.) JFM 30.0362.03
[20] S. U. Hwang, N. K. Kim, Y. Lee: On rings whose right annihilators are bounded. Glasg. Math. J. 51 (2009), 539-559. DOI 10.1017/S0017089509005163 | MR 2534006 | Zbl 1198.16001
[21] N. Jacobson: The Theory of Rings. Mathematical Surveys 2. AMS, New York (1943). DOI 10.1090/surv/002 | MR 0008601 | Zbl 0060.07302
[22] N. Jacobson: Structure of Rings. Colloquium Publication 37. AMS, Providence (1956). DOI 10.1090/coll/037 | MR 0081264 | Zbl 0073.02002
[23] I. Kaplansky: Commutative Rings. University of Chicago Press, Chicago (1974). MR 0345945 | Zbl 0296.13001
[24] W. F. Keigher: Adjunctions and comonads in differential algebra. Pac. J. Math. 59 (1975), 99-112. DOI 10.2140/pjm.1975.59.99 | MR 0392957 | Zbl 0327.12104
[25] W. F. Keigher: On the ring of Hurwitz series. Commun. Algebra 25 (1997), 1845-1859. DOI 10.1080/00927879708825957 | MR 1446134 | Zbl 0884.13013
[26] W. F. Keigher, F. L. Pritchard: Hurwitz series as formal functions. J. Pure Appl. Algebra 146 (2000), 291-304. DOI 10.1016/S0022-4049(98)00099-1 | MR 1742345 | Zbl 0978.12007
[27] J. Krempa: Some examples of reduced rings. Algebra Colloq. 3 (1996), 289-300. MR 1422968 | Zbl 0859.16019
[28] T. G. Lucas: Two annihilator conditions: Property (A) and (a.c.). Commun. Algebra 14 (1986), 557-580. DOI 10.1080/00927878608823325 | MR 0823354 | Zbl 0586.13004
[29] K. Paykan: Nilpotent elements of skew Hurwitz series rings. Rend. Circ. Mat. Palermo (2) 65 (2016), 451-458. DOI 10.1007/s12215-016-0245-y | MR 3571322 | Zbl 1353.16046
[30] K. Paykan: A study on skew Hurwitz series rings. Ric. Mat. 66 (2017), 383-393. DOI 10.1007/s11587-016-0305-9 | MR 3715907 | Zbl 1394.16050
[31] K. Paykan: Principally quasi-Baer skew Hurwitz series rings. Boll. Unione Mat. Ital. 10 (2017), 607-616. DOI 10.1007/s40574-016-0098-5 | MR 3736710 | Zbl 1381.16042
[32] Y. Quentel: Sur la compacité du spectre minimal d'un anneau. Bull. Soc. Math. Fr. 99 (1971), 265-272. (In French.) DOI 10.24033/bsmf.1720 | MR 0289496 | Zbl 0215.36803
[33] R. K. Sharma, A. B. Singh: Unified extensions of strongly reversible rings and links with other classic ring theoretic properties. J. Indian Math. Soc., New Ser. 85 (2018), 434-448. DOI 10.18311/jims/2018/20986 | MR 3816380 | Zbl 1463.16047
[34] R. K. Sharma, A. B. Singh: Zip property of skew Hurwitz series rings and modules. Serdica Math. J. 45 (2019), 35-54. MR 3971446
[35] R. K. Sharma, A. B. Singh: On rings with weak properties (A) and their extensions. J. Algebra Appl. 22 (2023), Article ID 2350112, 22 pages. DOI 10.1142/S0219498823501128 | MR 4556328 | Zbl 1536.16003
[36] R. K. Sharma, A. B. Singh: Skew Hurwitz series rings and modules with Beachy-Blair conditions. Kragujevac J. Math. 47 (2023), 511-521. MR 4654453 | Zbl 07891025
[37] R. K. Sharma, A. B. Singh: On a theorem of McCoy. Math. Bohem. 149 (2024), 27-38. DOI 10.21136/MB.2023.0031-22 | MR 4715554 | Zbl 07830541
[38] A. B. Singh, G. Sharma, F. Siddiqui: Classical properties of skew Hurwitz series rings. J. Algebra Relat. Top. 11 (2023), 105-115. DOI 10.22124/jart.2023.23391.1469 | MR 4699041 | Zbl 1538.16056
[39] E. J. Taft: Hurwitz invertibility of linearly recursive sequences. Combinatorics, Graph Theory, and Computing Congressus Numerantium 73. Utilitas Mathematica Publishing, Winnipeg (1990), 37-40. MR 1041834 | Zbl 0694.16006
[40] M. Zahiri, A. Moussavi, R. Mohammadi: On annihilator ideals in skew polynomial rings. Bull. Iran. Math. Soc. 43 (2017), 1017-1036. MR 3730622 | Zbl 1403.16025

Affiliations: Amit B. Singh (corresponding author), Department of Computer Science and Engineering, Jamia Hamdard (Deemed to be University), Mehrauli-Badarpur Rd, New Delhi-110062, India, e-mail: amit.bhooshan84@gmail.com, Deepa Arora, Department of Mathematics, Manav Rachna University, Faridabad 121004, Haryana, India, e-mail: deepa@mru.edu.in

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