On A New Class Of Ideals In Semirings
by
P. Mukhopadhyay
Department of Mathematics, Ramakrishna Mission Residential College, Narendrapur; affeliated to --University of Calcutta
Coauthors: M.K. Sen (Dept. of Pure Mathematics, University of Calcutta), Shamik Ghosh (Dept. of Mathematics; Jadavpur University)

The concept of p-ideal in a semiring was introduced by P. Mukhopadhyay and Shamik Ghosh in A new class of ideals in semirings; (South East Asian Bull. Math.  23,   (1999) pp.253-264;).

It is well-known that a ring R contains only one additive idempotent, namely the zero element. In a semiring S with additive idempotents, the set E⁺(S) forms an ideal of S, which is not necessarily a k-ideal. We consider the set P⁺(S)={x Î S : nx=(n+1)x for some n Î N} which consists of some additively periodic elements of S. Clearly, P⁺(R)={0} for any ring R. We note that P⁺(S) is an ideal of S, which is not necessarily a k-ideal but it has the following property: In a semiring S, let a Î P⁺(S) such that for some b Î S and some n Î N, a+nb = (n+1)b holds. Then b Î P⁺(S). This motivates us to define : "An ideal I of a semiring S is called a p-ideal if for some x Î S, n Î N, nx+a = (n+1)x and a Î I implies x Î I."

Clearly, in any halfring, every ideal is p-ideal. But not all p-ideals are k-ideals, as the ideal I=3Z⁺₀\{3} is not a k-ideal, in the halfring Z⁺₀ of all positive integers with zero. We also note that k-ideals are not p-ideals in general. Indeed, in the semiring (Z⁺, max, min),   I_n= { 1, 2, 3, ¼, n } is a k-ideal for any n Î Z⁺ but not a p-ideal.

In the above mentioned paper, we have developed p-simple semirings and then proving the existence of maximal p-ideals which are also k-ideals, the theory of p-semifield was developed and concept of p-primitivity was investigated.

We now define a new form of regularity in a semiring, which is compatible with the concept of p-ideal, as follows: A semiring S is called p- regular if for each a Î S, there exists some b Î S such that, na + aba  = (n+1)  a   for some n Î N. Examples are cited to justify that there exist p-regular semirings with n ¹ 1. It is shown that a semiring S is p-regular iff for every right p-ideal A and left p-ideal B in S, we have AÇB = [^AB], where [^AB] is the smallest p-ideal in S containing AB. We then obtain several chracterizations of p-regularity of a semiring in connection with p-ideals in it. Next we have defined the p-prime ideals, p-semiprime ideals and cited interesting examples of such classes and then fully p-prime semirings were introduced. Finally we have proved several results interlinking these concepts.

Date received: December 4, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cake-07.