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- Title
Reflexive ideals and reflexively closed subsets in rings.
- Authors
Kim, Sera; Kwak, Tai Keun; Lee, Chang Ik; Lee, Yang; Yun, Sang Jo
- Abstract
We continue the study of the reflexivity of ideals, introduced by Mason, and extend this notion to the subsets in rings. We first construct the smallest reflexive ideal containing S from any proper ideal S of any given ring R ; by which we can construct reflexive ideals but not semiprime in a kind of noncommutative ring. A subset T of a ring R is called reflexively closed if a R b ⊆ T for a , b ∈ R implies b R a ⊆ T , checking that a ring R is symmetric if and only if the right (left) annihilator of a is reflexively closed for any a ∈ R. We prove that the set of all nilpotent elements in a ring R is reflexively closed if and only if a R is nil for any nilpotent element a in R ; and that the Köthe's conjecture holds if and only if the union (sum) of the upper nilradical and any nil right ideal is reflexively closed. We provide another process to show that the set of all nilpotent elements of the polynomial ring over an NI ring need not be reflexively closed.
- Subjects
NONCOMMUTATIVE rings; ARTIN rings; POLYNOMIAL rings; MATRIX rings; REFLEXIVITY; RAMSEY numbers
- Publication
Journal of Algebra & Its Applications, 2023, Vol 22, Issue 5, p1
- ISSN
0219-4988
- Publication type
Article
- DOI
10.1142/S0219498823501062