Browsing by Author "Tamiru Abera"

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    On the Structural Properties of Ordered Weak Idempotent Rings
    (Addis Ababa University, 2024-08) Tamiru Abera; Kolluru Venkateswarlu; Yibeltal Yitayew
    The notion of weak idempotent rings is a new concept and has recently been introduced. It is a ring R of characteristic two, and a4 = a2 for each a in R. Very few studies have been done on its structural development and the structure of submaximal ideals of weak idempotent rings. So, there are a lot of gaps in its structural development. One of these gaps is the introduction of ordering in it. In this dissertation, we introduce the concept of partial ordering in a commutative weak idempotent ring R with unity and prove that the introduced partial ordering coincides with the one in Boolean rings whenever it is restricted to the idempotent parts of R. Besides, we introduce the concept of an atom in R and obtain certain results concerning an atom. Further, we study the structure of primary submaximal ideals. To achieve this, we de_ne partial ordering and an atom in a commutative weak idempotent ring with unity and use the concepts to obtain various results. Here we introduced the notion of partial ordering in a commutative weak idempotent ring R with unity, studied the role of an atom in the direct product decomposition of R and further on the structure of primary submaximal ideals of R. We recommend that researchers further study these in detail, shortly prove theorem 4.2.11, and extend the concepts to non-commutative weak idempotent rings with or without unity, and study its potential application in the real world.