Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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A NOTE ON MINIMAL PRIME IDEALS

  • Mohammadi, Rasul (Department of pure Mathematics Faculty of Mathematical Sciences Tarbiat Modares University) ;
  • Moussavi, Ahmad (Department of pure Mathematics Faculty of Mathematical Sciences Tarbiat Modares University) ;
  • Zahiri, Masoome (Department of Mathematics Faculty of Sciences Higher Education Center of Eghlid)
  • Received : 2016.06.28
  • Accepted : 2016.09.21
  • Published : 2017.07.31
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Abstract

Let R be a strongly 2-primal ring and I a proper ideal of R. Then there are only finitely many prime ideals minimal over I if and only if for every prime ideal P minimal over I, the ideal $P/{\sqrt{I}}$ of $R/{\sqrt{I}}$ is finitely generated if and only if the ring $R/{\sqrt{I}}$ satisfies the ACC on right annihilators. This result extends "D. D. Anderson, A note on minimal prime ideals, Proc. Amer. Math. Soc. 122 (1994), no. 1, 13-14." to large classes of noncommutative rings. It is also shown that, a 2-primal ring R only has finitely many minimal prime ideals if each minimal prime ideal of R is finitely generated. Examples are provided to illustrate our results.

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References

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