Valuations on ternary semirings. (English) Zbl 1495.16044
Summary: In the present study, we introduce a valuation of ternary semiring on an ordered abelian group. Motivated by the construction of valuation rings, we study some properties of ideals in ternary semiring arising in connection with the valuation map. We also explore ternary valuation semirings for a noncommuative ternary division semiring. We further consider the notion of convexity in a ternary semiring and how it is reflected in the valuation map.
MSC:
| 16Y60 | Semirings |
| 16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |
References:
| [1] | N. Bourbaki,Commutative Algebra, Chapters 1-7,Elements of Mathematics, Springer, 1998. · Zbl 0902.13001 |
| [2] | G. W. Chang and H. Kim,Divisibility properties of the semiring of ideals of an integral domain,Algebra Colloq.,27(3)(2020), 369-380. · Zbl 1448.13007 |
| [3] | M. Dehghanian and M. S. Modarres,Ternaryγ-homomorphisms and ternaryγderivations on ternary semigroups,J. Inequal. Appl.,34(2012). · Zbl 1275.39012 |
| [4] | V. R. Daddi and Y. S. Pawar,On completely regular ternary semiring,Novi Sad J. Math.,42(2)(2012), 1-7. · Zbl 1274.16070 |
| [5] | V. N. Dixit and S. Dewan,Minimal quasi-ideals in ternary semigroups,Indian J. Pure Appl. Math.,28(1997), 625-632. · Zbl 0885.20041 |
| [6] | T. K. Dutta and S. Kar,On regular ternary semirings,Advances in Algebra, Proceedings of the ICM Satellite Conference in Algebra and Related Topics, World Scientific (2003), 343-355. · Zbl 1041.16040 |
| [7] | T. K. Dutta and S. Kar,A note on regular ternary semirings,Kyungpook Math. J., 46(2006), 357-365. · Zbl 1122.16039 |
| [8] | T. K. Dutta, K. P. Shum and S. Mandal,Singular ideals of ternary semirings,Eur. J. Pure and Appl. Math.,5(2)(2012), 116-128. · Zbl 1389.16096 |
| [9] | T. K. Dutta, S. Kar and K. Das,Power ternary semirings,Afr. Mat.,26(2015), 1483-1494. · Zbl 1331.16039 |
| [10] | J. S. Golan,Semirings and their Applications,Springer Netherlands, 1999. · Zbl 0947.16034 |
| [11] | N. Gubareni,Valuation and discrete valuation rings,Scientific Research of the Institute of Mathematics and Computer Science,10(1)(2011), 61-70. |
| [12] | U. Hebisch and H. J. Weinert,Semirings: Algebraic Theory and Applications in Computer Science,Series in Algebra, World Scientific, 1998. · Zbl 0934.16046 |
| [13] | A. Iampan,Lateral ideals of ternary semigroups,Ukr. Math. Bull.,4(4)(2007), 525- 534. |
| [14] | S. Kar,On quasi-ideals and bi-ideals of ternary semirings,Int. J. Math. Math. Sci., 18(2005), 3015-3023. · Zbl 1094.16029 |
| [15] | S. Kar,Ideal theory in the ternary semiringZ−0,Bull. Malaya Math. Sci. Soc., (2)34(1)(2011), 69-77. · Zbl 1208.16039 |
| [16] | S. Kar and A. Shikari,Soft ternary semirings,Fuzzy Inf. Eng.,8(1)(2016), 1-15. |
| [17] | W. Krull,Allgemeine Bewertungstheorie,J. Reine Angew. Math.,167(1932), 160- 196. · Zbl 0004.09802 |
| [18] | W. Krull,Beitr¨age zur Arithmetik kommutativer Integrittsbereiche. VI. Der allgemeine Diskrimminantensatz,Unverzweigte Ringerweiterungen Math. Z.,45(1939), 1-19. · Zbl 0020.34003 |
| [19] | M. Manis,Valuations on a commutative ring,Proc. Amer. Math Soc.,20(1969), 193-198. · Zbl 0179.34201 |
| [20] | P. Nasehpour,Valuation Semirings,J. Algebra Appl.,17(4)(2018), 1850073 (23 pages). · Zbl 1385.16047 |
| [21] | O. F. G. Schilling,Noncommutative valuations,Bull. Amer. Math. Soc.,51(1945), 297-304. · Zbl 0061.05601 |
| [22] | G. Sheeja and S. Sri Bala,Congruences on ternary semigroups,Quasigroups Related Systems,20(2012), 113-124. · Zbl 1290.20066 |
| [23] | K. Valente,On orderings, valuations and0-primes of a commutative ring,Rocky Mt. J. Math.,19(3)(1989), 973-984. · Zbl 0706.13004 |
| [24] | O. Zariski and P. Samuel,Commutative Algebra Vol. II,D. Van Nostrand Company, Inc. (1960). · Zbl 0121.27801 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
