Weak Idempotent Rings
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Date
2020-11-11
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Addis Ababa University
Abstract
We introduce the notion of Weak idempotent ring (WIR, for short) which is a
ring of characteristic 2 and a4 = a2 for each a in the ring. We obtain certain
properties of this class. Further we provide examples of weak idempotent rings
to have more deeper insight in studying the structure of weak idempotent rings.
An equivalent de_nition for a commutative WIR with unity is given. Also we
obtain certain characterization theorems in terms of completely prime ideals and
left completely primary ideals. We introduce the concept of one sided completely
primary ideal and prove that a one sided completely primary ideal is completely
prime if the ideal contains all nilpotent elements of the ring. Also we prove that
every local weak idempotent ring with unity is primary ring and the intersection
of all primary ideals of a commutative weak idempotent ring with unity is the
zero ideal. We construct a partial synthesis of weak idempotent rings and develop
a subclass 2-Weak idempotent rings of the class of weak idempotent rings. We
investigate the structure of a weak idempotent ring with unity of 4 and 8 elements.
Further we prove that every proper ideal is nil whenever 0 and 1 are the only
idempotent elements of the weak idempotent ring with unity. We characterize the
semiprime and primary ideals of commutative weak idempotent rings with unity
and prove that the class weak idempotent rings satis_es the K othe's conjecture.
We study the structure of submaximal ideals in a commutative weak idempotent
ring with unity and show that every submaximal ideal of a commutative weak
idempotent ring with unity is either semiprime or primary. We prove that every
submaximal ideal of the product ring of two commutative WIRs with unity is
semiprime and the intersection of all submaximal ideals is the nilradical. We
make a study on the fraction of rings for commutative weak idempotent rings
with unity. Finally, We obtain certain properties concerning submaximal ideals
under homomorphic images.
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Keywords
Weak, Idempotent, Rings