AI Summary & Key Takeaways
AI InsightAs applications forthe additive results, we obtain some equivalent conditions of the n-strong Drazin invertibility of matrices over a ring.
📌 3 Key Takeaways
Abstract
Let R be a ring and n be a positive integer. In this paper, further results on the n-strong Drazin inverseare obtained in a ring. We prove that a ∈ R is n-strongly Drazin invertible if and only if a−an+1 is nilpotent. In termsof this characterization, the extensions of Cline’s formula and Jacobson’s lemma for this inverse are proved. Moreover,the n-strong Drazin invertibility for the sums of two elements is considered. We prove that a, b ∈ R are n-stronglyDrazin invertible if and only if a + b is n-strongly Drazin invertible, under the condition ab = 0. As applications forthe additive results, we obtain some equivalent conditions of the n-strong Drazin invertibility of matrices over a ring.