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
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.
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