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**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.

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 |

### Keywords:

sum of units; semicommutative rings; duo rings; nilpotent elements; idempotents; semicommutative \(\pi\)-regular rings; unit \(\pi\)-regular rings
Full Text:
DOI

### 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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