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Annihilator-stability and unique generation.
- Source :
-
Journal of Pure & Applied Algebra . Oct2017, Vol. 221 Issue 10, p2557-2572. 16p. - Publication Year :
- 2017
-
Abstract
- A ring R is said to be left uniquely generated if R a = R b in R implies that a = u b for some unit u in R . These rings have been of interest since Kaplansky introduced them in 1949 in his classic study of elementary divisors. Writing l ( b ) = { r ∈ R | r b = 0 } , a theorem of Canfell asserts that R is left uniquely generated if and only if, whenever R a + l ( b ) = R where a , b ∈ R , then a − u ∈ l ( b ) for some unit u in R . By analogy with the stable range 1 condition we call a ring with this property left annihilator-stable. In this paper we exploit this perspective on the left UG rings to construct new examples and derive new results. For example, writing J for the Jacobson radical, we show that a semiregular ring R is left annihilator-stable if and only if R / J is unit-regular, an analogue of Bass' theorem that semilocal rings have stable range 1. [ABSTRACT FROM AUTHOR]
- Subjects :
- *ALGEBRA
*MATHEMATICS
*EQUATIONS
*RING theory
Subjects
Details
- Language :
- English
- ISSN :
- 00224049
- Volume :
- 221
- Issue :
- 10
- Database :
- Academic Search Index
- Journal :
- Journal of Pure & Applied Algebra
- Publication Type :
- Academic Journal
- Accession number :
- 122880508
- Full Text :
- https://doi.org/10.1016/j.jpaa.2017.01.002