Date of Publication :13th March 2019
Abstract: A right near ring is an algebraic system with two binary operations such that (i) is a group - (not necessarily abelian) with 0 as its identity element, (ii) is a semigroup (we write for for all in ) and (iii) for all in . We say that is zero symmetric if for all in . is called an - near ring or an - near ring according as or for all . A subgroup of is called an N-subgroup if and an invariant Nsubgroup if, in addition, . An element in is said to be distributive, if for all and in ; is called distributively generated (d.g.), if the additive group of is generated by the multiplicative semigroup of distributive elements of . A near ring is defined to be right bipotent if for each in . In this paper, we have proved some more results on right bipotent near rings by using the concepts of - near ring ; subcommutativity ; regularity ; reduced property etc. It is proved that every right bipotent near ring is an - near ring and it is also - near ring if it is also subcommutative. Every regular near ring is central and reduced if it is right bipotent. Some special characterizations are obtained in such a way that, a reduced right bipotent near ring is a near field if and it is a division ring if it is dgnr.
Reference :
-
- R.BALAKRISHNAN and S.SURYANARAYANAN, P(r,m) Near Rings, Bull. Malasyian Math. Sc. Soc. (Second Series) 23 (2000), 117-130
- J.C.BEIDLEMAN, A note on regular near rings, J. Indian Math. Soc. 33 (1969), 207-210.
- V.R.CHANDRAN, On right bipotent rings, Abstract in Notices Amer. Math. Soc. Nov. (1970).
- J.R.CLAY, The near rings on groups of low order, Math. Z. 104 (1968), 364-371.
- GUNTER PILZ, Near rings, North Holland, Amsterdam, 1983. 6. A.FROHLICH, Distributively generated near rings (I Ideal Theory), Proc. London Math. Soc. 8 (1958), 76-94.
- H.E.HEATHERLY, Near rings without nilpotent elements, Pub. Math. Debrecen 20 (1973), 201-205.
- J.L.JAT and S.C.CHOUDHARY, On Left Bipotent Near-Rings, Proceedings of the Edinburgh Mathematical Society (1979), 22, 99 - 107©.
- K.KARTHY and P.DHEENA, On Unit Regular Near-Rings, Journal of the Indian Math., Soc. Vol.68, Nos 1-4: 2001, 239- 243.
- S.LIGH, On regular near rings, Math. Japonicae Ser (1) 15 (1970), 7-13.