ON STRONGLY REGULAR NEAR-SUBTRACTION SEMIGROUPS

• Dheena, P. (DEPARTMENT OF MATHEMATICS ANNAMALAI UNIVERSITY) ;
• Kumar, G. Satheesh (DEPARTMENT OF MATHEMATICS ANNAMALAI UNIVERSITY)
• Published : 2007.07.31
• 114 10

Abstract

In this paper we introduce the notion of strongly regular near-subtraction semigroups (right). We have shown that a near-subtraction semigroup X is strongly regular if and only if it is regular and without non zero nilpotent elements. We have also shown that in a strongly regular near-subtraction semigroup X, the following holds: (i) Xa is an ideal for every a $\in$ X (ii) If P is a prime ideal of X, then there exists no proper k-ideal M such that P $\subset$ M (iii) Every ideal I of X fulfills $I=I^2$.

Keywords

subtraction semigroup;near-subtraction semigroup;regular;strongly regular

References

1. J. C. Abbott, Sets, Lattices and Boolean Algebras, Allyn and Bacon, Boston, 1969
2. J. C. Beidleman, A note on Regular near-rings, J. Indian Math. Soc. 33 (1969), 207-210
3. J. R. Clay, The near-rings on groups of low order, Math. Z. 104 (1968), 364-371 https://doi.org/10.1007/BF01110428
4. K. H. Kim, On Subtraction Semiqroups, Scientiae Mathematicae Japonicae 62 (2005), no. 2, 273-280
5. Meldrum, Varieties and d.g. near-rings, Proc. Edinburgh Math. Soc. (series 1) 17 (1971), 271-274 https://doi.org/10.1017/S0013091500027000
6. G. Pilz, Near-rings, North-Holland, Amsterdam, 1983
7. E. H. Roh, K. H. Kim, and Jong Geol Lee, On Prime and Semiprime ideals in Subtraction Semiqroups, Scientiae Mathematicae Japonicae 61 (2005), no. 2, 259-266
8. B. M. Schein, Difference Semiqroups, Communications in algebra 20 (1992), 2153-2169 https://doi.org/10.1080/00927879208824453
9. B. Zelinka, Subtraction Semigroups, Math. Bohemica 120 (1995), 445-447
10. G. Mason, Strongly regular near-rings, Proc. Edinburgh Math. Soc. 23 (1980), 27-35 https://doi.org/10.1017/S0013091500003564