A NOTE ON STRONGLY *-CLEAN RINGS
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A *-ring R is called (strongly) *-clean if every element of R is the sum of a projection and a unit (which commute with each other). In this note, some properties of *-clean rings are considered. In particular, a new class of *-clean rings which called strongly π-*-regular are introduced. It is shown that R is strongly π-*-regular if and only if R is π-regular and every idempotent of R is a projection if and only if R/J(R) is strongly regular with J(R) nil, and every idempotent of R/J(R) is lifted to a central projection of R. In addition, the stable range conditions of *-clean rings are discussed, and equivalent conditions among *-rings related to *-cleanness are obtained.
Abstract
A *-ring R is called (strongly) *-clean if every element of R is the sum of a projection and a unit (which commute with each other). In this note, some properties of *-clean rings are considered. In particular, a new class of *-clean rings which called strongly π-*-regular are introduced. It is shown that R is strongly π-*-regular if and only if R is π-regular and every idempotent of R is a projection if and only if R/J(R) is strongly regular with J(R) nil, and every idempotent of R/J(R) is lifted to a central projection of R. In addition, the stable range conditions of *-clean rings are discussed, and equivalent conditions among *-rings related to *-cleanness are obtained.
참고문헌 (16)
- A. Badawi, "On abelian -regular rings", Comm. Algebra, 25(4), pp.1009~1021 (1997)
- S. K. Berberian, "Baer *-Rings", Grundlehren der Mathematischen Wissenschaften, 195 (1972), Springer-Verlag
- K. P. S. Bhaskara Rao, "The Theory of Generalized Inverses over Commutative Rings" (2002), Taylor & Francis
- G. Borooah, "Strongly clean matrix rings over commutative local rings", J. Pure Appl. Algebra, 212(1), pp.281~296 (2008)
- V. P. Camillo, "Exchange rings, units and idempotents", Comm. Algebra, 22(12), pp.4737~4749 (1994)
- H. Chen, "Rings with many idempotents", Int. J. Math. Math. Sci, 22(3), pp.547~558 (1999)
- J. Chen, "Two questions of L. Vaˇs on *-clean rings", Bull. Aust. Math. Soc, 88(3), pp.499~505 (2013)
- J. Chen, "Group rings in which every element is uniquely the sum of a unit and an idempotent", J. Algebra, 306(2), pp.453~460 (2006)
- J. Chen, "When is the 2×2 matrix ring over a commutative local ring strongly clean?", J. Algebra, 301(1), pp.280~293 (2006)
- A. Y. M. Chin, "On strongly π-regular group rings", Southeast Asian Bull. Math, 26(3), pp.387~390 (2002)
- C. Li, "On strongly *-clean rings", J. Algebra Appl, 10(6), pp.1363~1370 (2011)
- W. K. Nicholson, "Lifting idempotents and exchange rings", Trans. Amer. Math. Soc, 229, pp.269~278 (1977)
- W. K. Nicholson, "Strongly clean rings and Fitting’s lemma", Comm. Algebra, 27(8), pp.3583~3592 (1999)
- W. K. Nicholson, "Rings in which elements are uniquely the sum of an idempotent and a unit", Glasg. Math. J, 46(2), pp.227~236 (2004)
- L. Vaˇs, "*-Clean rings: some clean and almost clean Baer *-rings and von Neumann algebras", J. Algebra, 324(12), pp.3388~3400 (2010)
- Z. Wang, "Rings of idempotent stable rang one", Algebr. Represent. Theory, 15(1), pp.195~200 (2012)