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- Title
On (1,r)-ideals of commutative rings.
- Authors
Anebri, Adam; Khalfi, Abdelhaq El; Mahdou, Najib
- Abstract
Let R be a commutative ring with nonzero identity. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of r -ideals. A proper ideal I of R is said to be a (1 , r) -ideal if whenever nonunit elements a , b , c ∈ R and a b c ∈ I , then a b ∈ I or c ∈ Z (R). Some basic properties of (1 , r) -ideals are studied. For instance, we give a method to construct (1 , r) -ideals that are not r -ideals. Among other things, it is shown that if R admits a (1 , r) -ideal that is not an r -ideal, then R is a local ring. Also, we provide a necessary and sufficient condition in term of (1 , r) -ideals for a ring to be a total quotient ring and we determinate the (1 , r) -ideals of a chained ring. Finally, we give an idea about some (1 , r) -ideals of the localization of rings, the trivial ring extensions and the power series rings to construct nontrivial and original examples of (1 , r) -ideals.
- Subjects
COMMUTATIVE rings; QUOTIENT rings; LOCAL rings (Algebra); POWER series; RING theory
- Publication
Journal of Algebra & Its Applications, 2024, Vol 23, Issue 2, p1
- ISSN
0219-4988
- Publication type
Article
- DOI
10.1142/S0219498824500233