Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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WHEN IS C(X) AN EM-RING?

  • Abuosba, Emad (Department of Mathematics The University of Jordan) ;
  • Atassi, Isaaf (Department of Mathematics The University of Jordan)
  • Received : 2020.12.04
  • Accepted : 2021.04.22
  • Published : 2022.01.31
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Abstract

A commutative ring with unity R is called an EM-ring if for any finitely generated ideal I there exist a in R and a finitely generated ideal J with Ann(J) = 0 and I = aJ. In this article it is proved that C(X) is an EM-ring if and only if for each U ∈ Coz (X), and each g ∈ C* (U) there is V ∈ Coz (X) such that U ⊆ V, ${\bar{V}}=X$, and g is continuously extendable on V. Such a space is called an EM-space. It is shown that EM-spaces include a large class of spaces as F-spaces and cozero complemented spaces. It is proved among other results that X is an EM-space if and only if the Stone-Čech compactification of X is.

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Acknowledgement

The authors are very grateful for the referee's efforts and valuable comments, which improved the article and brought it this way.

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