hello
im looking for a equivalent conditions for strpngly \pi regular rings. a ring R is said to be strongly \pi regular if for each x in R there exist a y in R such that x^n = (x ^(n+1))y.
do you know any equivalent conditions for such rings?
There are some such conditions in "Strongly $\pi$-regular Rings" by Azumaya (https://projecteuclid.org/download/pdf_1/euclid.hokmj/1530842562). You can find a number of references simply by googling "strongly \pi regular rings".
thank you but im looking for ring equivalent for example we know that a ring R is von Neumann regullar an orthogonally finite if and only if R is semisimple.
This is an important question.
Consider: Every weakly nil clean ring of bounded index and every weakly nil clean PI-ring is strongly π-regular, observed in
Peter Danchev and Janez Šter
Generalizing π-regular rings
https://arxiv.org/pdf/1412.4359.pdf
It is observed that the center of a weakly nil clean ring is (strongly) π-regular, which extends the analogous result of McCoy [24] for π-regular rings (p. 2).
24] N. H. McCoy. Generalized regular rings. Bull. Amer. Math. Soc., 45(2):175–178, 1939.
See Prop. 3.8, Prop. 3.10, Cor. 3.10, Cor. 3.11 and Lemma 3.9, p. 6.
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