1 results
USING IDEALS TO PROVIDE A UNIFIED APPROACH TO UNIQUELY CLEAN RINGS
- Part of
-
- Journal:
- Journal of the Australian Mathematical Society / Volume 96 / Issue 2 / April 2014
- Published online by Cambridge University Press:
- 01 April 2014, pp. 258-274
- Print publication:
- April 2014
-
- Article
-
- You have access
- Export citation
-
In this article, we introduce the notion of the uniquely
$I$-clean ring and show that, if 
$R$ is a ring and 
$I$ is an ideal of 
$R$ then 
$R$ is uniquely 
$I$-clean if and only if (
$R/ I$ is Boolean and idempotents lift uniquely modulo 
$I$) if and only if (for each 
$a\in R$ there exists a central idempotent 
$e\in R$ such that 
$e- a\in I$ and 
$I$ is idempotent-free). We examine when ideal extension is uniquely clean relative to an ideal. Also we obtain conditions on a ring 
$R$ and an ideal 
$I$ of 
$R$ under which uniquely 
$I$-clean rings coincide with uniquely clean rings. Further we prove that a ring 
$R$ is uniquely nil-clean if and only if (
$N(R)$ is an ideal of 
$R$ and 
$R$ is uniquely 
$N(R)$-clean) if and only if 
$R$ is both uniquely clean and nil-clean if and only if (
$R$ is an abelian exchange ring with 
$J(R)$ nil and every quasiregular element is uniquely clean). We also show that 
$R$ is a uniquely clean ring such that every prime ideal of 
$R$ is maximal if and only if 
$R$ is uniquely nil-clean ring and 
$N(R)= {\mathrm{Nil} }_{\ast } (R)$.
