NOTE ON n-(1, r)-IDEALS OF COMMUTATIVE RINGS
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In this paper, we introduce and study n-(1, r)-ideals of commutative rings with nonzero identity. Let R be a ring and n be a positive integer. A proper ideal I of R is called an n-(1, r)-ideal if whenever nonunit elements x₁, ..., xₙ ∈ R and x₁⋯xₙ₊₁ ∈ I, then x₁x₂⋯xₙ ∈ I or xₙ₊₁ ∈ Z(R). Various examples and characterizations of n-(1, r)-ideals are given. For example, we prove that if R admits an n-(1, r)-ideal that is not an (n−1)-(1, r)-ideal, then R is a local ring. We provide an example of an n-(1, r)-ideal that is not an (n−1)-(1, r)-ideal. In addition, we give a description of n-(1, r)-ideals in chained rings. Finally, we study the transfer of n-(1, r)-ideals in the localization of rings, the power series rings and the trivial ring extension.
Abstract
In this paper, we introduce and study n-(1, r)-ideals of commutative rings with nonzero identity. Let R be a ring and n be a positive integer. A proper ideal I of R is called an n-(1, r)-ideal if whenever nonunit elements x₁, ..., xₙ ∈ R and x₁⋯xₙ₊₁ ∈ I, then x₁x₂⋯xₙ ∈ I or xₙ₊₁ ∈ Z(R). Various examples and characterizations of n-(1, r)-ideals are given. For example, we prove that if R admits an n-(1, r)-ideal that is not an (n−1)-(1, r)-ideal, then R is a local ring. We provide an example of an n-(1, r)-ideal that is not an (n−1)-(1, r)-ideal. In addition, we give a description of n-(1, r)-ideals in chained rings. Finally, we study the transfer of n-(1, r)-ideals in the localization of rings, the power series rings and the trivial ring extension.
참고문헌 (16)
- D. F. Anderson, "On n-Absorbing ideals of commutative rings", Comm. Algebra, 39(5), pp.1646~1672 (2011)
- D. D. Anderson, "Idealization of a module", J. Commut. Algebra, 1(1), pp.3~56 (2009)
- A. Anebri, "r)-ideals of commutative rings", J. Algebra Appl, 23(2), pp.2450023~ (2024)
- A. Anebri, "On modules satisfying the descending chain condition on r-submodules", Comm. Algebra, 50(1), pp.383~391 (2022)
- A. Badawi, "On 1-absorbing primary ideals of commutative rings", J. Algebra Appl, 19(6), pp.2050111~ (2020)
- J. W. Brewer, "Power Series over Commutative Rings" (1981), Marcel Dekker, Inc
- D. E. Dobbs, "Trivial extensions satisfying certain valuation-like properties", Comm. Algebra, 47(5), pp.2060~2077 (2019)
- S. Glaz, "Commutative Coherent Rings" (1989), Springer-Verlag
- J. A. Huckaba, "Commutative Rings with Zero Divisors" (1988), Dekker
- S. Kabbaj, "Matlis’ semi-regularity and semi-coherence in trivial ring extensions: a survey", Moroccan Journal of Algebra and Geometry with Applications, 1(1), pp.1~17 (2022)
- S. Kabbaj, "Trivial extensions defined by coherent-like conditions", Comm. Algebra, 32(10), pp.3937~3953 (2004)
- I. Kaplansky, "Commutative Rings" (1974), Univ. Chicago Press
- R. Mohamadian, "r-Ideals in commutative rings", Turk. J. Math, 39, pp.733~749 (2015)
- N. Mahdou, "Weakly prime ideals issued from an amalgamated algebra", Hacet. J. Math. Stat, 49(3), pp.1159~1167 (2020)
- G. Ulucak, "On n-1-absorbing prime ideals", J. Algebra Appl, 22(10), pp.2350207~ (2023)
- A. Yassine, "On 1-absorbing prime ideals of commutative rings", J. Algebra Appl, 20(10), pp.2150175~ (2021)