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Commutative nil-clean and \(\pi\)-regular group rings. (English) Zbl 1488.16111

Summary: Our first achievement in the present paper is to give a more simple and transparent proof of a result due to W. Wm. McGovern et al. in [J. Algebra Appl. 14, No. 6, Article ID 1550094, 5 p. (2015; Zbl 1325.16024)] concerning commutative nil-clean group rings and P. V. Danchev and W. Wm. McGovern in [J. Algebra 425, 410–422 (2015; Zbl 1316.16028)] concerning commutative weakly nil-clean group rings, respectively. Our method used for proof allows us to considerably extend both theorems to commutative \(\pi\)-regular group rings, thus generalizing results due to M. Auslander in [Proc. Am. Math. Soc. 8, 658–664 (1957; Zbl 0079.26703)], I. G. Connel in [Can. J. Math. 15, 650–685 (1963; Zbl 0121.03502)] and S. V. Mihovski in [“On the strongly regular group rings”, Bull. Inst. Math. Acad. Bulg. Sci. 14, 67–72 (1971)] concerned with regular and strongly regular group rings, respectively. Our results are also paralleling to these by A. Y. M. Chin and H. V. Chen in [Southeast Asian Bull. Math. 26, No. 3, 387–390 (2002; Zbl 1032.16010)].

MSC:

16U99 Conditions on elements
16S34 Group rings
16U60 Units, groups of units (associative rings and algebras)
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