Periodica Mathematica Hungarlca Vol. 17 (1), (1986), pp. 13--20

D-STRONG AND ALMOST D-STRONG NEAR-RINGS

A. K. GOYAL (Udaipur)

Abstract

In this paper, D-strong and almost D-strong near-rings have been defined. It has been proved that if R is a D-strong S-near ring, then prime ideals, strictly prime ideals and completely prime ideals coincide. Also if R is a D-strong near-ring with iden- tity, then every maximal right ideal becomes a maximal ideal and moreover every 2- primitive near-ring becomes a near-field. Several properties, chain conditions and structure theorems have also been discussed.

Introduction

In this paper, we have generalized some of the results obtained for rings by Wong [12]. Corresponding to the prime and strictly prime ideals in near-rings, we have defined D-strong and almost D-strong near-rings. It has been shown that a regular near-ring having all idempotents central in R is a D-strong and hence almost D-strong near-ring. If R is a D-strong S-near ring, then it has been shown that prime ideals, strictly prime ideals and com- pletely prime ideals coincide and g(R) = H(R) ~ ~(R), where H(R) is the intersection of all strictly prime ideals of R. Also if R is a D-strong near-ring with identity, then every maximal right ideal becomes a maximal ideal and moreover every 2-primitive near-ring becomes a near-field. Some structure theorems have also been discussed.

Preliminaries

Throughout R will denote a zero-symmetric left near-ring, i.e., R -- Ro in the sense of Pilz [10]. A near-ring R is said to be an S-near ring if a EaR for all a E R.

Most of the parts of this paper are included in author's doctoral dissertation a~ Sukhadia University Udaipur (1983). The author expresses his gratitude to Dr. S. C. CHOUDHARY for his kind guidance. AMS (MOS) subject classifications (1970). Primary 16A96; Secondary 16A56. Key word8 and phrases. D-strong , almost D-strong, prime, ideal, strictly prime ideal, completely prime and completely semi-prime ideals, regular, strongly regular, 2,prhnitive near-ring, G-radical, .near-field. -

Akad~mlai K~uI6, Budapest D. Reidel,Dordre~ht 14 GOYAL: D-STRONG N]~AR-RINGS

An additive subgroup H of R is called an R-subgroup if

HR---- {hrlhE H, rE R} c_H. A normal subgroup B of R is said to be a right ideal of R if

(rl + b)r2 -- rlr2 E B for all b E B and r 1, rg. E R. Every right ideal is an R-subgroup but not con- versely. For definitions of R-modules, d.g. near-rings, nearfields and subdirect product we refer to Pilz [10]. The following definitions can be found in [9]. An ideal P of R is said to be strictly prime ideal if for any R-subgroups A and B of R such that AB c_ P, then we have either A ___ P or B ___ P. Clearly a strictly prime ideal is prime but not conversely. Also if R is an S-near ring, then P is a strictly prime ideal of R iff for a, b~R, aRb~P~aEPorbEP. An idealPofR is said to be strictly semiprime if for any R-subgroup A of R such that A 2 _ P, then we have A c_ P. Every strictly prime ideal is strictly semiprime. A near- ring R is called s*-prime (s*-semiprime) [strictly prime and strictly semi- prime in the sense of Oswald [9] but not strictly prime in the sense of Pilz ([10], p. 68)] if the (0) ideal is strictly prime (strictly semiprime) ideal in R. The definition for completely prime and completely semiprime ideals are the same as in ring case [6]. Clearly a completely prime ideal is strictly prime but not conversely. A near-ring R is said to be simple if it contains no proper two-sided ideals. R is said to be strongly semisimple (semisimple in the sense of Blackett [1]) if it contains no non-zero nilpotent R-subgroups and satisfies d.c.e, on R-subgroups [2]. An R-module Mn is said to be R-simple in the sense of Pilz ([10], p. 19) (irreducible in the sense of Choudhary [2]) if it contains no proper R-subgroups. A right ideal B of R is called R-simple if as an R-mod- ule it is R-simple. A near-ring R is said to be regular if for each a E R, there exists x E R such that a ~ axa. R is called right strongly regular if for each a E R, there exists x E R such that a = a2x [8]. In a zero-symmetric case, every right strongly regular near-ring is regular and contains no non-zero nilpotent elements [8]. For arbitrary subsets A and B or R, (A:B)----(xERIBxc_A} is usually called a noetherian quotient. If A is an ideal and B an R'subgroup of R, then it is easy to see that (A : B) is an ideal of R. Unless otherwise stated, the number/~ will be E {0, 1, 2}. A right ideal B of R is called strictly maximal if B is maximal as an R-subgroup. A near-