On Duo and Strict-Duo Near-Rings
by
S.C. Choudhary
Dept of Mathematics, M.L. Sukhadia University, Udaipur, India

As now it is known that a near-ring satisfies fewer axioms than those in a ring (except: commutativity of addition and one of the distributive laws). A near-ring N is called duo if every one-sided ideal is also two-sided; and strict-duo if every N-subgroup is also closed from the other side. In a ring case these two coincide but not in a near-ring. The follwing results are of importance:

(1) If N is a duo near-ring with left identity and J₂(N) = 0, then N is isomorphic to a subdirect sum of near-fields. (J₂(N) is Jacobson like radical in near-rings).

(2) If N is a duo near-ring with left identity, then the following are equivalent: (a) N is a finite direct sum of ideals which are also near-fields; (b) N is strongly (and strictly) semisimple; (c) N satisfies the dcc on ideals and J₂(N) = 0; (d) N a finite direct sum of ideals which are also 2-primitive near-rings.

(3) If N is a strict-duo regular near-ring, then each ideal A = L(A) = U(A) = \surdA = {x in N | xⁿ in A for some n} where L(A),  (U(A)) is the intersection of all the prime (s-prime) ideals of N containing A.

(4) Each duo regular near-ring is G-semisimple, i.e. G(N) = 0, where G(N) is the G-radical (Brown-McCoy radical) in near-rings.

Date received: February 16, 1999