Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON STRONGLY 1-ABSORBING PRIMARY IDEALS OF COMMUTATIVE RINGS

  • Almahdi, Fuad Ali Ahmed (Department of Mathematics Faculty of Sciences King Khalid University) ;
  • Bouba, El Mehdi (Department of Mathematics Faculty of Science University Moulay Ismail) ;
  • Koam, Ali N.A. (Department of Mathematics College of Science Jazan University)
  • Received : 2019.10.01
  • Accepted : 2020.03.26
  • Published : 2020.09.30
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Abstract

Let R be a commutative ring with 1 ≠ 0. In this paper, we introduce a subclass of the class of 1-absorbing primary ideals called the class of strongly 1-absorbing primary ideals. A proper ideal I of R is called strongly 1-absorbing primary if whenever nonunit elements a, b, c ∈ R and abc ∈ I, then ab ∈ I or c ∈ ${\sqrt{0}}$. Firstly, we investigate basic properties of strongly 1-absorbing primary ideals. Hence, we use strongly 1-absorbing primary ideals to characterize rings with exactly one prime ideal (the UN-rings) and local rings with exactly one non maximal prime ideal. Many other results are given to disclose the relations between this new concept and others that already exist. Namely, the prime ideals, the primary ideals and the 1-absorbing primary ideals. In the end of this paper, we give an idea about some strongly 1-absorbing primary ideals of the quotient rings, the polynomial rings, and the power series rings.

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References

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