대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제38권1호
  • /
  • Pages.39-46
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  • 2023
  • /
  • 1225-1763(pISSN)
  • /
  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
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ON THE 2-ABSORBING SUBMODULES AND ZERO-DIVISOR GRAPH OF EQUIVALENCE CLASSES OF ZERO DIVISORS

  • Shiroyeh Payrovi (Department of Mathematics Imam Khomeini International University) ;
  • Yasaman Sadatrasul (Department of Mathematics Imam Khomeini International University)
  • 투고 : 2021.11.21
  • 심사 : 2022.07.28
  • 발행 : 2023.01.31
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초록

Let R be a commutative ring, M be a Noetherian R-module, and N a 2-absorbing submodule of M such that r(N :R M) = 𝖕 is a prime ideal of R. The main result of the paper states that if N = Q1 ∩ ⋯ ∩ Qn with r(Qi :R M) = 𝖕i, for i = 1, . . . , n, is a minimal primary decomposition of N, then the following statements are true. (i) 𝖕 = 𝖕k for some 1 ≤ k ≤ n. (ii) For each j = 1, . . . , n there exists mj ∈ M such that 𝖕j = (N :R mj). (iii) For each i, j = 1, . . . , n either 𝖕i ⊆ 𝖕j or 𝖕j ⊆ 𝖕i. Let ΓE(M) denote the zero-divisor graph of equivalence classes of zero divisors of M. It is shown that {Q1∩ ⋯ ∩Qn-1, Q1∩ ⋯ ∩Qn-2, . . . , Q1} is an independent subset of V (ΓE(M)), whenever the zero submodule of M is a 2-absorbing submodule and Q1 ∩ ⋯ ∩ Qn = 0 is its minimal primary decomposition. Furthermore, it is proved that ΓE(M)[(0 :R M)], the induced subgraph of ΓE(M) by (0 :R M), is complete.

키워드

과제정보

The authors would like to thank the referee for a careful reading of our paper and insightful comments which saved us from several errors.

참고문헌

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