The Journal of the Indian Mathematical Society

Open Access Open Access  Restricted Access Subscription Access
Open Access Open Access Open Access  Restricted Access Restricted Access Subscription Access

On K-regular Additive Ternary Semirings


Affiliations
1 Department of Mathematics, M. J. College, Jalgaon - 425002, India
2 Department of Mathematics, ACS College, Dharangaon - 425 105, India
     

   Subscribe/Renew Journal


We introduce the concepts of a k-regular and a k-invertible additive ternary semiring. We show that (i) If I is a k-regular ideal of an additive ternary semiring S and J is any ideal of S, then I ? J is a k-regular ideal of S; (ii) If S is an additively idempotent, commutative additive ternary semiring and x ? S, then M (x) is a commutative additive ternary monoid of (S, +); (iii) An additively idempotent additive ternary semiring S is k-regular if and only if S is k-invertible; (iv) Let S be an additively and lateral cancellative additive ternary semiring. If a, b ? S, then V (a) and V (b) are either disjoint or equal.


Keywords

Additive Ternary Semiring; Additively Idempotent Additive Ternary Semiring; K-Regular Additive Ternary Semiring; K-Invertible Additive Ternary Semiring.
Subscription Login to verify subscription
User
Notifications
Font Size


  • M. R. Adhikari, M. K. Sen and H. J. Weinert, On k-regular semrings, Bull. Cal. Math. Soc., 88 (1996), 141–144.

  • S. Bourne, The Jacobson radical of a semiring, Proc. Nat. Acad. Sci. U.S.A., 37 (1951), 163–170.

  • J. N. Chaudhari, H. P. Bendale and K. J. Ingale, Regular ternary semirings, J. Adv. Res. Pure Math., 4(3) (2012), 68–76.

  • J. N. Chaudhari and K. J. Ingale, On k-regular semirings, J. Indian Math. Soc., 82 (3–4), (2015), 1–11.

  • T. K. Dutta and S. Kar, On regular ternary semirings, Advances in Algebra, Proceedings of the ICM Satellite Conference in Algebra and Related Topics, Worid Scientific (2003), 343–355.

  • Shamik Ghosh, A note on regularity in matrix semirings, Kyungpook Math. J., 44 (2004), 1–4.

  • J. S. Golan, Semiring and Their Applications, Kluwer Academic publisher Dordrecht, 1999.

  • Vishnu Gupta and J. N. Chaudhari, On right -regular semirings, Sarajevo J. Math., 2(14) (2006), 3–9.

  • W. G. Lister, Ternary rings, Trans. Amer. Math. Soc., 154 (1971) 37–55.

  • A. Pop, Remarks on embedding theorems of (m, n)-semirings, Bul. Stiint. Univ. Baia Mare Ser. B, Mathematica–Informatica 16 (2000), 297–302.

  • M. K. Sen and A. K. Bhuniya, Completely k-regular semirings, Bull. Cal. Math. Soc., 97(5) (2005), 455–466.

  • H. S. Vandiver, Note on a simple type of algebra in which cancellation law of addition does not hold, Bull. Amer. Math. Soc., 40 (1934), 914–920.


Abstract Views: 175

PDF Views: 0




  • On K-regular Additive Ternary Semirings

Abstract Views: 175  |  PDF Views: 0

Authors

Kunal Julal Ingale
Department of Mathematics, M. J. College, Jalgaon - 425002, India
Hemant Premraj Bendale
Department of Mathematics, M. J. College, Jalgaon - 425002, India
Dipak Ravindra Bonde
Department of Mathematics, ACS College, Dharangaon - 425 105, India
Jayprakash Ninu Chaudhari
Department of Mathematics, M. J. College, Jalgaon - 425002, India

Abstract


We introduce the concepts of a k-regular and a k-invertible additive ternary semiring. We show that (i) If I is a k-regular ideal of an additive ternary semiring S and J is any ideal of S, then I ? J is a k-regular ideal of S; (ii) If S is an additively idempotent, commutative additive ternary semiring and x ? S, then M (x) is a commutative additive ternary monoid of (S, +); (iii) An additively idempotent additive ternary semiring S is k-regular if and only if S is k-invertible; (iv) Let S be an additively and lateral cancellative additive ternary semiring. If a, b ? S, then V (a) and V (b) are either disjoint or equal.


Keywords


Additive Ternary Semiring; Additively Idempotent Additive Ternary Semiring; K-Regular Additive Ternary Semiring; K-Invertible Additive Ternary Semiring.

References





DOI: https://doi.org/10.18311/jims%2F2022%2F29309