Recently, a new type of generalized inverse called the n-strong Drazin inverse was introduced by Mosić in the setting of rings. Namely, let R be a ring and n be a positive integer, an element x ∈ R is called the n-strong Drazin inverse of a ∈ R if it satisfies xax = x, ax = xa and an − ax ∈ Rnil. The main aim of this paper is to consider some equivalent characterizations for the n-strong Drazin invertibility in a ring. Firstly, we give an equivalent definition of the n-strong Drazin inverse, that is, x is the n-strong Drazin inverse of a if and only if xax = x, xa = ax and a − an+2x ∈ Rnil. Also, we obtain some existence criteria for this inverse by means of idempotents. In particular, the n-strong Drazin invertibility of the product paq are investigated, where a is regular and p,q are arbitrary elements in a ring.

中文翻译：

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环中 n-强 Drazin 可逆性的表征

最近，一种新的广义逆称为n- 强烈的 Drazin inverse 是 Mosić 在戒指的设置中引入的。即，让R成为一个戒指和n是一个正整数，一个元素X ∈ R被称为n- 强 Drazin 逆一种 ∈ R如果满足X一种X = X,一种X = X一种和一种n - 一种X ∈ R零. 本文的主要目的是考虑一些等效的表征n- 环中的强 Drazin 可逆性。首先，我们给出一个等价的定义n-强Drazin逆，即，X是个n- 强 Drazin 逆一种当且仅当X一种X = X,X一种 = 一种X和一种 - 一种n+2X ∈ R零. 此外，我们通过幂等性获得了这个逆的一些存在标准。特别是，n-产品的强Drazin可逆性p一种q被调查，其中一种是规律的并且p,q是环中的任意元素。