In this paper the authors prove the following result:
Let \(R\) be a ring of bounded index \(n\) for which there exists an element whose nonzero \(n-1\) power is von Neumann regular. We have:

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if \(R\) is prime, then it is isomorphic to a matrix ring over a unital domain;

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if \(R\) is indecomposable, then it is isomorphic to a matrix ring over a unital ring without nilpotent elements;

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if \(R\) is von Neumann regular, then it is isomorphic to a matrix ring over an abelian regular ring, and, therefore,

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if \(R\) is prime and von Neumann regular, it is isomorphic to a matrix ring over a division ring.

The key tool of this result is a Jordan decomposition of any nilpotent element of index \(n\) whose \(n-1\) power is von Neumann regular. This elements can be decomposed into an orthogonal sum of a Jordan block of size \(n\) over a corner of the original ring, and a nilpotent part of lower index. This result generalizes one of [K. I. Beidar et al. [Commun. Algebra 32, No. 9, 3543–3562 (2004; Zbl 1074.16005)], where the condition of being von Neumann regular was required for the whole ring.