| Title:
|
On a theorem of McCoy (English) |
| Author:
|
Sharma, Rajendra Kumar |
| Author:
|
Singh, Amit B. |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
149 |
| Issue:
|
1 |
| Year:
|
2024 |
| Pages:
|
27-38 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study McCoy's theorem to the skew Hurwitz series ring $({\rm HR}, \omega )$ for some different classes of rings such as: semiprime rings, APP rings and skew Hurwitz serieswise quasi-Armendariz rings. Moreover, we establish an equivalence relationship between a right zip ring and its skew Hurwitz series ring in case when a ring $R$ satisfies McCoy's theorem of skew Hurwitz series. (English) |
| Keyword:
|
skew Hurwitz series ring |
| Keyword:
|
$\omega $-compatible ring |
| Keyword:
|
skew Hurwitz serieswise |
| Keyword:
|
quasi-Armendariz rings |
| Keyword:
|
zip ring |
| Keyword:
|
APP ring |
| MSC:
|
16S10 |
| MSC:
|
16S85 |
| MSC:
|
16U80 |
| idZBL:
|
Zbl 07830541 |
| idMR:
|
MR4715554 |
| DOI:
|
10.21136/MB.2023.0031-22 |
| . |
| Date available:
|
2024-03-13T10:16:50Z |
| Last updated:
|
2024-12-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152290 |
| . |
| Reference:
|
[1] Ahmadi, M., Moussavi, A., Nourozi, V.: On skew Hurwitz serieswise Armendariz rings.Asian-Eur. J. Math. 7 (2014), Article ID 1450036, 19 pages. Zbl 1308.16033, MR 3257511, 10.1142/S1793557114500363 |
| Reference:
|
[2] Annin, S.: Associated primes over Ore extensions rings.J. Algera Appl. 3 (2004), 193-205. Zbl 1060.16029, MR 2069261, 10.1142/S0219498804000782 |
| Reference:
|
[3] Armendariz, E. P.: A note on extensions of Baer and P.P.-rings.J. Aust. Math. Soc. 18 (1974), 470-473. Zbl 0292.16009, MR 0366979, 10.1017/S1446788700029190 |
| Reference:
|
[4] Beachy, J. A., Blair, W. D.: Rings whose faithful left ideals are cofaithful.Pac. J. Math. 58 (1975), 1-13. Zbl 0309.16004, MR 0393092, 10.2140/pjm.1975.58.1 |
| Reference:
|
[5] Benhissi, A., Koja, F.: Basic properties of Hurwitz series rings.Ric. Mat. 61 (2012), 255-273. Zbl 1318.13034, MR 3000659, 10.1007/s11587-012-0128-2 |
| Reference:
|
[6] Birkenmeier, G. F., Kim, J. Y., Park, J. K.: On quasi-Baer rings.Algebra and Its Applications Contemporary Mathematics 259. AMS, Providence (2000), 67-92. Zbl 0974.16006, MR 1778495, 10.1090/conm/259 |
| Reference:
|
[7] Cedó, F.: Zip rings and Mal'cev domains.Commun. Algebra 19 (1991), 1983-1991. Zbl 0733.16007, MR 1121118, 10.1080/00927879108824242 |
| Reference:
|
[8] Cortes, W.: Skew polynomial extensions over zip rings.Int. J. Math. Math. Sci. 2008 (2008), Article ID 496720, 9 pages. Zbl 1159.16021, MR 2393011, 10.1155/2008/496720 |
| Reference:
|
[9] Faith, C.: Rings with zero intersection property on annihilators: Zip rings.Publ. Mat., Barc. 33 (1989), 329-338. Zbl 0702.16015, MR 1030970, 10.5565/PUBLMAT_33289_09 |
| Reference:
|
[10] Faith, C.: Annihilator ideals, associated primes and Kasch-McCoy commutative rings.Commun. Algebra 19 (1991), 1867-1892. Zbl 0729.16015, MR 1121111, 10.1080/00927879108824235 |
| Reference:
|
[11] Fields, D. E.: Zero divisors and nilpotent elements in power series rings.Proc. Am. Math. Soc. 27 (1971), 427-433. Zbl 0219.13023, MR 0271100, 10.1090/S0002-9939-1971-0271100-6 |
| Reference:
|
[12] Fliess, M.: Sur divers produits de séries formelles.Bull. Soc. Math. Fr. 102 (1974), 181-191 French. Zbl 0313.13021, MR 0354647, 10.24033/bsmf.1777 |
| Reference:
|
[13] Gilmer, R., Grams, A., Parker, T.: Zero divisors in power series rings.J. Reine Angew. Math. 278/279 (1975), 145-164. Zbl 0309.13009, MR 0387274, 10.1515/crll.1975.278-279.145 |
| Reference:
|
[14] Hashemi, E., Moussavi, A.: Polynomial extensions of quasi-Baer rings.Acta. Math. Hung. 107 (2005), 207-224. Zbl 1081.16032, MR 2148584, 10.1007/s10474-005-0191-1 |
| Reference:
|
[15] Hassanein, A. M.: Clean rings of skew Hurwitz series.Matematiche 62 (2007), 47-54. Zbl 1150.16029, MR 2389111 |
| Reference:
|
[16] Hirano, Y.: On annihilator ideals of a polynomial ring over a noncommutative ring.J. Pure Appl. Algebra 168 (2002), 45-52. Zbl 1007.16020, MR 1879930, 10.1016/S0022-4049(01)00053-6 |
| Reference:
|
[17] Hong, C. Y., Kim, N. K., Kwak, T. K., Lee, Y.: Extensions of zip rings.J. Pure Appl. Algebra 195 (2005), 231-242. Zbl 1071.16020, MR 2114273, 10.1016/j.jpaa.2004.08.025 |
| Reference:
|
[18] Hong, C. Y., Kim, N. K., Lee, Y.: Extensions of McCoy's theorem.Glasg. Math. J. 52 (2010), 155-159. Zbl 1195.16026, MR 2587825, 10.1017/S0017089509990243 |
| Reference:
|
[19] Jones, L. G., Weiner, L.: Advanced problems and solutions: Solutions 4419. Concerning a theorem of McCoy.Am. Math. Mon. 59 (1952), 336-337. MR 0048481, 10.1080/00029890.1952.11988133 |
| Reference:
|
[20] Keigher, W. F.: Adjunctions and comonads in differential algebra.Pac. J. Math. 59 (1975), 99-112. Zbl 0327.12104, MR 0392957, 10.2140/pjm.1975.59.99 |
| Reference:
|
[21] Keigher, W. F.: On the ring of Hurwitz series.Commun. Algebra 25 (1997), 1845-1859. Zbl 0884.13013, MR 1446134, 10.1080/00927879708825957 |
| Reference:
|
[22] Keigher, W. F., Pritchard, F. L.: Hurwitz series as formal functions.J. Pure Appl. Algebra 146 (2000), 291-304. Zbl 0978.12007, MR 1742345, 10.1016/S0022-4049(98)00099-1 |
| Reference:
|
[23] Krempa, J.: Some examples of reduced rings.Algebra Colloq. 3 (1996), 289-300. Zbl 0859.16019, MR 1422968 |
| Reference:
|
[24] Leroy, A., Matczuk, J.: Zip property of certain ring extensions.J. Pure Appl. Algebra 220 (2016), 335-345. Zbl 1334.16019, MR 3393464, 10.1016/j.jpaa.2015.06.015 |
| Reference:
|
[25] Liu, Z., Zhao, R.: A generalization of PP-rings and p.q.-Baer rings.Glasg. Math. J. 48 (2006), 217-229. Zbl 1110.16003, MR 2256973, 10.1017/S0017089506003016 |
| Reference:
|
[26] McCoy, N. H.: Remarks on divisors of zero.Am. Math. Mon. 49 (1942), 286-295. Zbl 0060.07703, MR 0006150, 10.1080/00029890.1942.11991226 |
| Reference:
|
[27] McCoy, N. H.: Annihilators in polynomial rings.Am. Math. Mon. 64 (1957), 28-29. Zbl 0077.25903, MR 0082486, 10.1080/00029890.1957.11988927 |
| Reference:
|
[28] Paykan, K.: Nilpotent elements of skew Hurwitz series rings.Rend. Circ. Mat. Palermo (2) 65 (2016), 451-458. Zbl 1353.16046, MR 3571322, 10.1007/s12215-016-0245-y |
| Reference:
|
[29] Paykan, K.: A study on skew Hurwitz series rings.Ric. Mat. 66 (2017), 383-393. Zbl 1394.16050, MR 3715907, 10.1007/s11587-016-0305-9 |
| Reference:
|
[30] Paykan, K.: Principally quasi-Baer skew Hurwitz series rings.Bull. Unione Mat. Ital. 10 (2017), 607-616. Zbl 1381.16042, MR 3736710, 10.1007/s40574-016-0098-5 |
| Reference:
|
[31] Rege, M. B., Chhawchharia, S.: Armendariz rings.Proc. Japan Acad., Ser. A 73 (1997), 14-17. Zbl 0960.16038, MR 1442245, 10.3792/pjaa.73.14 |
| Reference:
|
[32] Sharma, R. K., Singh, A. B.: Unified extensions of strongly reversible rings and links with other classic ring theoretic properties.J. Indian Math. Soc., New Ser. 85 (2018), 434-448. Zbl 1463.16047, MR 3816380, 10.18311/jims/2018/20986 |
| Reference:
|
[33] Sharma, R. K., Singh, A. B.: Zip property of skew Hurwitz series rings and modules.Serdica Math. J. 45 (2019), 35-54. MR 3971446 |
| Reference:
|
[34] Sharma, R. K., Singh, A. B.: Skew Hurwitz series rings and modules with Beachy-Blair conditions.Kragujevac J. Math. 47 (2023), 511-521. MR 4654453, 10.46793/KgJMat2304.511S |
| Reference:
|
[35] Singh, A. B., Dixit, V. N.: Unification of extensions of zip rings.Acta Univ. Sapientiae, Math. 4 (2012), 168-181. Zbl 1295.16015, MR 3123260 |
| Reference:
|
[36] Stenström, B.: Rings of Quotients: An Introduction to Methods of Ring Theory.Die Grundlehren der mathematischen Wissenschaften 217. Springer, Berlin (1975). Zbl 0296.16001, MR 0389953, 10.1007/978-3-642-66066-5 |
| Reference:
|
[37] Taft, E. J.: Hurwitz invertibility of linearly recursive sequences.Combinatorics, Graph Theory, and Computing Congressus Numerantium 73. Utilitas Mathematica, Winnipeg (1990), 37-40. Zbl 0694.16006, MR 1041834 |
| Reference:
|
[38] Tominaga, H.: On $s$-unital rings.Math. J. Okayama Univ. 18 (1976), 117-134. Zbl 0335.16020, MR 0453801 |
| Reference:
|
[39] Zelmanowitz, J. M.: The finite intersection property on annihilator right ideals.Proc. Am. Math. Soc. 57 (1976), 213-216. Zbl 0333.16014, MR 0419512, 10.1090/S0002-9939-1976-0419512-6 |
| . |
