Geodesic
On strongly r-ideals on commutative rings and some related graphs
Filomat, Tome 37 (2023) no. 15, p. 5027

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DOI

A proper ideal I of a ring R is called an r-ideal if, whenever x, y ∈ R with xy ∈ I, we have x ∈ I or y ∈ Z(R) [R. Mohamadian, r-ideals in commutative rings, Turkish J. Math. 39(5) (2015),733-749]. In this article, we are interested in a subclass of the class of r-ideals which we call the class of strongly r-ideals. A proper ideal I of a ring R is called a strongly r-ideal if, whenever x, y ∈ R with xy ∈ I, we have x ∈ I or y ∈ Z(I). First, we give a basic study of this new concept which includes, among others, characterizations, properties and examples. After that, we use the introduced concept to characterize rings for which the diameter of the zero-divisor graph is less than or equal to two, rings for which the annihilator graph is complete, and rings for which the zero-annihilator graph is empty.
DOI : 10.2298/FIL2315027A
Classification : 13A15, 13E05, 13F20
Keywords: (Strongly) r-ideals, Total ring of fractions, The zero-divisor graph, McCoy rings 
Khaled Alhazmy; Fuad Ali Ahmed Almahdi; El Mehdi Bouba; Mohammed Tamekkante. On strongly r-ideals on commutative rings and some related graphs. Filomat, Tome 37 (2023) no. 15, p. 5027 . doi: 10.2298/FIL2315027A
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     title = {On strongly r-ideals on commutative rings and some related graphs},
     journal = {Filomat},
     pages = {5027 },
     year = {2023},
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     doi = {10.2298/FIL2315027A},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2315027A/}
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