×

Exchange ideals with all idempotents central. (English) Zbl 1292.16008

An ideal \(I\) of a unital ring \(R\) is called an exchange ideal provided \(I\) is a non-unital exchange ring in the sense of P. Ara [J. Algebra 197, No. 2, 409-423 (1997; Zbl 0890.16003)]. Ara proved that \(I\) is an exchange ideal of \(R\) if and only if for any \(x\in I\), there is an idempotent \(e\in xI\) (equivalently, \(e\in xR\)) such that \(1-e\in (1-x)R\).
In the present paper, conditions are developed under which \(I\) is an Abelian exchange ideal, meaning that in addition to the exchange condition, all idempotents in \(I\) are central. The main characterization of this condition is that \(I\) is an Abelian exchange ideal of \(R\) if and only if every element of \(I\) is the sum of a central idempotent of \(R\) and a unit of \(R\). This leads to a number of alternative characterizations, which are applied to certain ring constructions. In particular, suppose a ring \(S\) (possibly non-unital) supports a suitably compatible \(R\)-\(R\)-bimodule structure. Then \(I(R;S):=R\oplus S\) has a natural ring structure, and the author characterizes when ideals of \(I(R;S)\) of the form \(K\oplus S\), where \(K\) is an ideal of \(R\), are Abelian exchange ideals. Corollaries include that the power series ring \(R[[x]]\) is an Abelian exchange ring if and only if \(R\) is an Abelian exchange ring. Finally, Abelian exchange rings are related to uniquely clean ideals, where an ideal \(I\) of \(R\) is said to be uniquely clean provided that for any \(a\in I\), there is a unique idempotent \(e\in 1+I\) such that \(a-e\) is a unit. It is proved that every uniquely clean ideal is an Abelian exchange ideal. Also, if \(I\) is an ideal of \(R\) containing \(2\cdot1_R\), then \(I\) is uniquely clean if and only if it is an Abelian exchange ideal and \(R/M\cong\mathbb Z_2\) for all maximal ideals \(M\) of \(R\) not containing \(I\).

MSC:

16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16U80 Generalizations of commutativity (associative rings and algebras)
16U60 Units, groups of units (associative rings and algebras)
16D25 Ideals in associative algebras

Citations:

Zbl 0890.16003
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chen, Algebra Colloq. 6 pp 45– (1999)
[2] DOI: 10.1080/00927870500261538 · Zbl 1100.16007 · doi:10.1080/00927870500261538
[3] DOI: 10.1016/j.jalgebra.2006.08.012 · Zbl 1110.16025 · doi:10.1016/j.jalgebra.2006.08.012
[4] DOI: 10.4134/CKMS.2002.17.3.409 · Zbl 1104.16302 · doi:10.4134/CKMS.2002.17.3.409
[5] DOI: 10.1017/S0017089508004370 · Zbl 1159.16006 · doi:10.1017/S0017089508004370
[6] DOI: 10.1016/j.jalgebra.2006.10.032 · Zbl 1117.16017 · doi:10.1016/j.jalgebra.2006.10.032
[7] Nicholson, NJ pp 181– (2005)
[8] DOI: 10.1017/S0017089504001727 · Zbl 1057.16007 · doi:10.1017/S0017089504001727
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.