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**Commutative weakly nil clean unital rings.**
*(English)*
Zbl 1316.16028

In this short and interesting paper the authors, generalizing the notions of weakly clean ring and nil clean ring, define weakly nil clean rings and investigate their properties for commutative rings. They prove that every weakly nil clean ring is zero-dimensional and hence a clean ring. They prove that the class of all weakly nil clean rings is closed under taking homomorphic images but not closed under taking finite direct products. They prove that a weakly nil clean ring with 2 nilpotent is a clean ring. Their main theorem asserts that a ring \(R\) is weakly nil clean if and only if \(R/N(R)\) is isomorphic to either a Boolean ring or \(Z_3\) or a product of two such rings.

Next they study weakly nil clean group rings. They completely characterize weakly nil clean group rings. They prove that a group ring \(R[G]\) (for \(G\) abelian) is weakly nil clean if and only if exactly one of the following three conditions is satisfied. (i) \(R\) is nil clean and \(G\) is a nontrivial torsion 2-group; (ii) \(R/N(R)\) is isomorphic to \(Z_3\) and \(G\) is a nontrivial torsion 3-group; (iii) \(R\) is weakly nil clean and \(G\) is trivial.

Finally, the authors study extension rings \(R[X]\), \(R(X)\), \(R\langle X\rangle\) and \(R[[X]]\). They prove that none of these rings is weakly nil clean. The authors conclude the paper with some open problems.

Next they study weakly nil clean group rings. They completely characterize weakly nil clean group rings. They prove that a group ring \(R[G]\) (for \(G\) abelian) is weakly nil clean if and only if exactly one of the following three conditions is satisfied. (i) \(R\) is nil clean and \(G\) is a nontrivial torsion 2-group; (ii) \(R/N(R)\) is isomorphic to \(Z_3\) and \(G\) is a nontrivial torsion 3-group; (iii) \(R\) is weakly nil clean and \(G\) is trivial.

Finally, the authors study extension rings \(R[X]\), \(R(X)\), \(R\langle X\rangle\) and \(R[[X]]\). They prove that none of these rings is weakly nil clean. The authors conclude the paper with some open problems.

Reviewer: Veereshwar A. Hiremath (Dharwad)

### MSC:

16U60 | Units, groups of units (associative rings and algebras) |

13A99 | General commutative ring theory |

16S34 | Group rings |

16N40 | Nil and nilpotent radicals, sets, ideals, associative rings |

16U80 | Generalizations of commutativity (associative rings and algebras) |

### Keywords:

Boolean rings; weakly nil clean rings; polynomial rings; weakly nil clean group rings; Nagata rings
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\textit{P. V. Danchev} and \textit{W. Wm. McGovern}, J. Algebra 425, 410--422 (2015; Zbl 1316.16028)

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### References:

[1] | Ahn, M. S.; Anderson, D. D., Weakly clean rings and almost Clean rings, Rocky Mountain J. Math., 36, 783-798, (2006) · Zbl 1131.13301 |

[2] | Anderson, D. D., Multiplication ideals, multiplication rings, and the ring \(R(X)\), Canad. J. Math., 28, 760-768, (1976) · Zbl 0343.13009 |

[3] | Anderson, D. D.; Anderson, D. F.; Markanda, R., The rings \(R(X)\) and \(R \langle X \rangle\), J. Algebra, 95, 96-115, (1985) · Zbl 0621.13008 |

[4] | Anderson, D. D.; Camilo, V. P., Commutative rings whose elements are a sum of a unit and idempotent, Comm. Algebra, 30, 3327-3336, (2002) · Zbl 1083.13501 |

[5] | Breaz, S.; Calugareanu, G.; Danchev, P.; Micu, T., Nil-Clean matrix rings, Linear Algebra Appl., 439, 3115-3119, (2013) · Zbl 1355.16023 |

[6] | Chen, H., On uniquely Clean rings, Comm. Algebra, 39, 189-198, (2010) · Zbl 1251.16027 |

[7] | Chen, H., On strongly nil Clean matrices, Comm. Algebra, 41, 1074-1086, (2013) · Zbl 1286.16025 |

[8] | Chen, H.; Kose, H.; Kurtulmaz, Y., Strongly P-Clean rings and matrices, Int. Electron. J. Algebra, 15, 115-131, (2014) · Zbl 1296.16039 |

[9] | Chin, A. Y.M.; Qua, K. T., A note on weakly Clean rings, Acta Math. Hungar., 132, 113-116, (2011) · Zbl 1232.16022 |

[10] | Cohn, P. M., An introduction to ring theory, (2000), Springer-Verlag Berlin, New York and London · Zbl 1009.16001 |

[11] | Diesl, A. J., Nil Clean rings, J. Algebra, 383, 197-211, (2013) · Zbl 1296.16016 |

[12] | Fields, D. E., Zero divisors and nilpotent elements in power series rings, Proc. Amer. Math. Soc., 27, 427-433, (1971) · Zbl 0219.13023 |

[13] | Gilmer, R., Commutative semigroup rings, Chicago Lectures in Mathematics, (1984), University of Chicago Press Chicago, IL · Zbl 0566.20050 |

[14] | Han, J.; Nicholson, W. K., Extensions of Clean rings, Comm. Algebra, 29, 2589-2595, (2001) · Zbl 0989.16015 |

[15] | Huckaba, J. A., Commutative rings with zero divisors, (1988), Marcel Dekker New York and London · Zbl 0637.13001 |

[16] | Immormino, N., Clean rings and Clean group rings, (2013), Bowling Green State University, Ph.D. Dissertation |

[17] | Immormino, N.; McGovern, W. Wm., Examples of Clean commutative group rings, J. Algebra, 405, 168-178, (2014) · Zbl 1305.16022 |

[18] | Kocan, M. T.; Lee, T.-K.; Zhou, Y., When is every matrix over a division ring a sum of an idempotent and a nilpotent?, Linear Algebra Appl., 450, 7-12, (2014) · Zbl 1303.15016 |

[19] | McGovern, W. Wm., Neat rings, J. Pure Appl. Algebra, 205, 243-265, (2006) · Zbl 1095.13025 |

[20] | McGovern, W. Wm., A characterization of commutative Clean rings, Int. J. Math. Game Theory Algebra, 15, 403-413, (2006) · Zbl 1130.13304 |

[21] | McGovern, W. Wm.; Raja, S.; Sharp, A., Commutative nil Clean group rings, J. Algebra Appl., 14, (2015) · Zbl 1325.16024 |

[22] | McGovern, W. Wm.; Richman, F., When \(R(X)\) and \(R \langle X \rangle\) are Clean: a constructive treatment, Comm. Algebra, 43, (2015), in press · Zbl 1320.13008 |

[23] | Nicholson, W. K., Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., 229, 269-278, (1977) · Zbl 0352.16006 |

[24] | Nicholson, W. K.; Zhou, Y., Rings in which elements are uniquely the sum of an idempotent and a unit, Glasg. Math. J., 46, 227-236, (2004) · Zbl 1057.16007 |

[25] | Nicholson, W. K.; Zhou, Y., Clean general rings, J. Algebra, 291, 297-311, (2005) · Zbl 1084.16023 |

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