EXCHANGE ELEMENTS IN RINGS, AND THE EQUATION XA - BX = I

被引：12

作者：

Khurana, Dinesh ^{[1
]}

Lam, T. Y. ^{[2
]}

Nielsen, Pace P. ^{[3
]}

机构：

[1] Panjab Univ, Dept Math, Chandigarh 160014, India

[2] Univ Calif Berkeley, Dept Math, Berkeley, CA 94720 USA

[3] Brigham Young Univ, Dept Math, Provo, UT 84602 USA

来源：

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY | 2017年 / 369卷 / 01期

关键词：

Idempotents; exchange elements; suitable elements; regular elements; clean elements; linear and quadratic equations; suitable rings; STRONGLY CLEAN RINGS; MODULES;

D O I：

10.1090/tran6652

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

The equation XA-BX = I has been well studied in ring theory, operator theory, linear algebra, and other branches of mathematics. In this paper, we show that, in the case where B-2 = B, the study of XA-BX = I in a noncommutative ring R leads to several new ways to view and to work with the exchange (or "suitable") elements in R in the sense of Nicholson. For any exchange element A is an element of R, we show that the set of idempotents E is an element of R such that E is an element of RA and I-E is an element of R(I-A) is naturally parametrized by the roots of a certain left-right symmetric "exchange polynomial" associated with A. From the new viewpoints on exchange elements developed in this paper, the classes of clean and strongly clean elements in rings can also be better understood.

引用

页码：495 / 516

页数：22

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