## On semicommutative $$\pi$$-regular rings.(English)Zbl 0803.16008

An element $$a$$ in a ring $$R$$ with identity is called semicommutative if for every $$b \in R$$ there are $$r,s \in R$$ such that $$ab = ra$$ and $$ba = as$$. If every element of $$R$$ is semicommutative, then $$R$$ is called a semicommutative ring. Note that semicommutative rings are also called duo rings. A ring $$R$$ is called $$\pi$$-regular (resp. unit $$\pi$$-regular) if for every $$x \in R$$ there exists a positive integer $$n$$ (depending on $$x$$) and an element $$y \in R$$ (resp. a unit element $$u \in R$$) such that $$x^ n = x^ n yx^ n$$ (resp. $$x^ n = x^ n ux^ n$$).
For a semicommutative ring $$R$$, it is shown that the set $$\text{Nil}(R)$$ of all nilpotent elements forms a two-sided ideal, and $$R$$ is $$\pi$$- regular if and only if $$R/\text{Nil}(R)$$ is von Neumann regular. This result was also proved by Y. Hirano [Math. J. Okayama Univ. 20, 141-149 (1978; Zbl 0394.16011)]. After proving that all idempotents of a semicommutative ring are central, it is shown that all semicommutative $$\pi$$-regular rings are unit $$\pi$$-regular rings. Moreover, it is shown that if 2 is a unit in a semicommutative $$\pi$$-regular ring $$R$$, then every element of $$R$$ is a sum of two units in $$R$$, which is parallel to a result of J. W. Fisher and R. L. Snider [J. Algebra 42, 363- 368 (1976; Zbl 0335.16014)] for the case of $$\pi$$-regular rings whose primitive factor rings are Artinian.
Reviewer: J.K.Park (Pusan)

### MSC:

 16E50 von Neumann regular rings and generalizations (associative algebraic aspects) 16U80 Generalizations of commutativity (associative rings and algebras) 16N40 Nil and nilpotent radicals, sets, ideals, associative rings

### Citations:

Zbl 0394.16011; Zbl 0335.16014
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### References:

 [1] Goodearl, K. R. 1979. ”Von Neumann Regular Rings”. Pitman Publishing Limited. · Zbl 0411.16007 [2] Lam, T. Y. 1991. ”A first Course in Noncommutative Rings”. New York: Springer Verlag. Inc · Zbl 0728.16001
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