Browsing by Author "Biadiglign Asmare"

Now showing 1 - 1 of 1
  • No Thumbnail Available
    Item
    Weak Idempotent Nil-neat Rings
    (Addis Ababa University, 2024-08-31) Biadiglign Asmare; Kolluru Venkateswarlu; Tilahun Abebaw
    We introduce the concept of a weak idempotent nil-clean ring which is a generalization of weakly nil-clean ring. We give certain characterizations for a weak idempotent nil-clean ring in terms of the Jacobson radical and nil-radical. In addition to this, we prove that n n upper (lower) triangular matrix over a ring R is weak idempotent nil-clean if and only if so is R. We introduce the concept of a strongly weak idempotent nil-clean ring which is a generalization of a strongly weakly nil clean ring. We characterize strongly weak idempotent nil-clean rings in terms of the set of nil-potent elements, homomorphic images, and Jacobson radicals. Moreover, we give necessary and sufficient conditions of a strongly weak idempotent nil-clean ring in relation to periodic rings, and also we give a characterization between strongly weak idempotent nil-clean rings and strongly -regular rings and strongly clean rings element wise. Furthermore, we prove that a strongly weak idempotent nil-clean ring R with 2 2 J(R) satisfies nil-involution property. We define the concept of a weak idempotent nil-neat ring which is the generalization of a weakly nil-neat ring. We characterize reduced weak idempotent nil-clean rings. Also, we give a characterization of weak idempotent nil-neat rings in terms of semiprime ideals, maximal ideals and Jacobson radicals. Moreover, we prove that every nonzero prime ideal of a strongly weak idempotent nil-clean ring is maximal. Finally, we investigate the condition for which the group ring R[G] becomes a weak idempotent nil-clean ring and a weak idempotent nil-neat ring.