| Title:
|
Invertible ideals and Gaussian semirings (English) |
| Author:
|
Ghalandarzadeh, Shaban |
| Author:
|
Nasehpour, Peyman |
| Author:
|
Razavi, Rafieh |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
53 |
| Issue:
|
3 |
| Year:
|
2017 |
| Pages:
|
179-192 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In the first section, we introduce the notions of fractional and invertible ideals of semirings and characterize invertible ideals of a semidomain. In section two, we define Prüfer semirings and characterize them in terms of valuation semirings. In this section, we also characterize Prüfer semirings in terms of some identities over its ideals such as $(I + J)(I \cap J) = IJ$ for all ideals $I$, $J$ of $S$. In the third section, we give a semiring version for the Gilmer-Tsang Theorem, which states that for a suitable family of semirings, the concepts of Prüfer and Gaussian semirings are equivalent. At last, we end this paper by giving a plenty of examples for proper Gaussian and Prüfer semirings. (English) |
| Keyword:
|
semiring |
| Keyword:
|
semiring polynomials |
| Keyword:
|
Gaussian semiring |
| Keyword:
|
cancellation ideals |
| Keyword:
|
invertible ideals |
| MSC:
|
06D75 |
| MSC:
|
13B25 |
| MSC:
|
13F25 |
| MSC:
|
16Y60 |
| idZBL:
|
Zbl 06819524 |
| idMR:
|
MR3708771 |
| DOI:
|
10.5817/AM2017-3-179 |
| . |
| Date available:
|
2017-09-13T09:35:35Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146883 |
| . |
| Reference:
|
[1] Arnold, J.T., Gilmer, R.: On the content of polynomials.Proc. Amer. Math. Soc. 40 (1) (1970), 556–562. MR 0252360, 10.1090/S0002-9939-1970-0252360-3 |
| Reference:
|
[2] Bazzoni, S., Glaz, S.: Gaussian properties of total rings of quotients.J. Algebra 310 (1) (2007), 180–193. Zbl 1118.13020, MR 2307788, 10.1016/j.jalgebra.2007.01.004 |
| Reference:
|
[3] Bourne, S.: The Jacobson radical of a semiring.Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 163–170. Zbl 0042.03201, MR 0041827, 10.1073/pnas.37.3.163 |
| Reference:
|
[4] Dale, L., Pitts, J.D.: Euclidean and Gaussian semirings.Kyungpook Math. J. 18 (1978), 17–22. Zbl 0399.16018, MR 0491842 |
| Reference:
|
[5] Dedekind, R.: Supplement XI to P.G. Lejeune Dirichlet: Vorlesung über Zahlentheorie 4 Aufl..ch. Über die Theorie der ganzen algebraiscen Zahlen, Druck und Verlag, Braunschweig, 1894. MR 0237283 |
| Reference:
|
[6] Eilenberg, S.: Automata, Languages, and Machines.vol. A, Academic Press, New York, 1974. Zbl 0317.94045, MR 0530382 |
| Reference:
|
[7] El Bashir, R., Hurt, J., Jančařík, A., Kepka, T.: Simple commutative semirings.J. Algebra 236 (2001), 277–306. Zbl 0976.16034, MR 1808355, 10.1006/jabr.2000.8483 |
| Reference:
|
[8] Gilmer, R.: Some applications of the Hilfsatz von Dedekind-Mertens.Math. Scand. 20 (1967), 240–244. MR 0236159, 10.7146/math.scand.a-10833 |
| Reference:
|
[9] Gilmer, R.: Multiplicative Ideal Theory.Marcel Dekker, New York, 1972. Zbl 0248.13001, MR 0427289 |
| Reference:
|
[10] Glazek, K.: A guide to the literature on semirings and their applications in mathematics and information sciences.Kluwer Academic Publishers, Dordrecht, 2002. Zbl 1072.16040, MR 2007485 |
| Reference:
|
[11] Golan, J.S.: Semirings and Their Applications.Kluwer Academic Publishers, Dordrecht, 1999. Zbl 0947.16034, MR 1746739 |
| Reference:
|
[12] Golan, J.S.: Power Algebras over Semirings: with Applications in Mathematics and Computer Science.vol. 488, Springer, 1999. Zbl 0947.16035, MR 1730722 |
| Reference:
|
[13] Hebisch, U., Weinert, H.J.: On Euclidean semirings.Kyungpook Math. J. 27 (1987), 61–88. Zbl 0645.16025, MR 0922411 |
| Reference:
|
[14] Hebisch, U., Weinert, H.J.: Semirings - Algebraic Theory and Applications in Computer Science.World Scientific, Singapore, 1998. Zbl 0934.16046, MR 1704233 |
| Reference:
|
[15] Kim, C.B.: A note on the localization in semirings.J. Sci. Inst. Kookmin Univ. 3 (1985), 13–19. |
| Reference:
|
[16] Kurosh, A.G.: Lectures in General Algebra.Pergamon Press, Oxford, 1965, translated by A. Swinfen. Zbl 0123.00101, MR 0179235 |
| Reference:
|
[17] LaGrassa, S.: Semirings: Ideals and Polynomials.Ph.D. thesis, University of Iowa, 1995. MR 2692760 |
| Reference:
|
[18] Larsen, M.D., McCarthy, P.J.: Multiplicative Theory of Ideals.Academic Press, New York, 1971. Zbl 0237.13002, MR 0414528 |
| Reference:
|
[19] Naoum, A.G., Mijbass, A.S.: Weak cancellation modules.Kyungpook Math. J. 37 (1997), 73–82. Zbl 0882.13002, MR 1454770 |
| Reference:
|
[20] Nasehpour, P.: On the content of polynomials over semirings and its applications.J. Algebra Appl. 15 (5) (2016), 32, 1650088. MR 3479447, 10.1142/S0219498816500882 |
| Reference:
|
[21] Nasehpour, P.: Valuation semirings.J. Algebra Appl. 16 (11) (2018), 23, 1850073, arXiv:1509.03354. |
| Reference:
|
[22] Nasehpour, P., Yassemi, S.: $M$-cancellation ideals.Kyungpook Math. J. 40 (2000), 259–263. Zbl 1020.13002, MR 1803117 |
| Reference:
|
[23] Noronha Galvão, M.L.: Ideals in the semiring $\mathbb{N}$.Portugal. Math. 37 (1978), 231–235. MR 0620304 |
| Reference:
|
[24] Prüfer, H.: Untersuchungen über Teilbarkeitseigenschaften in Körpern.J. Reine Angew. Math. 168 (1932), 1–36. Zbl 0004.34001, MR 1581355 |
| Reference:
|
[25] Smith, F.: Some remarks on multiplication modules.Arch. Math. (Basel) 50 (1988), 223–235. Zbl 0615.13003, MR 0933916, 10.1007/BF01187738 |
| Reference:
|
[27] Vandiver, H.S.: Note on a simple type of algebra in which cancellation law of addition does not hold.Bull. Amer. Math. Soc. 40 (1934), 914–920. MR 1562999, 10.1090/S0002-9904-1934-06003-8 |
| . |
