Fully idempotent near‐rings and sheaf representations

Javed Ahsan,Gordon Mason

doi:10.1155/s0161171298000192

Abstract

Fully idempotent near‐rings are defined and characterized which yields information on the lattice of ideals of fully idempotent rings and near‐rings. The space of prime ideals is topologized and a sheaf representation is given for a class of fully idempotent near‐rings which includes strongly regular near‐rings.

Highlights

AND PRELIMINARIESFor basic terminology and results on near-rings, see [10] and [11]
If I is an ideal of R, I is called prime if whenever A, B are ideals with AB C_I either A C_I or B if_I; I is completely prime’if ab I => a I or b 6 I; I is semiprime if A C_I => A C_I; I is irreducible if A n B I => A I or
A ring R is fully idempotent if each ideal I of R is idempotent, i.e. if/ 12. Several characterizations of these rings were given by Courter [7] and they play an important role in the study of

Summary

INTRODUCTION

Throughout this paper R will denote a zero-symmetric right near-ring. If I is an ideal of R, I is called prime if whenever A, B are ideals with AB C_ I either A C_ I or B if_ I; I is completely prime’if ab I => a I or b 6 I; I is semiprime if A C_ I => A C_ I; I is irreducible Note that unlike the situation in rings a prime ideal I need not have the property aRb C_ I => a I or b E I.

FULLY IDEMPOTENT NEAR-RINGS

STRONGLY IDEMPOTNT AND STRONGLY RULAR NAR-RINGS