Common properties of some subrings of $\mathbb{R}^X$

Document Type : Research Paper

Author

Department of Pure Mathematics, Faculty of Mathematics, Yazd University,Yazd, Iran

Abstract

For a nonempty topological space $X$, the ring of all real-valued functions on $X$ with pointwise addition and multiplication is denoted by $F(X)$ and contonuous members of $F(X)$ is denoted by $C(X)$. The collection of all pointwise limit functions of sequences in $C(X)$ is denoted by $B_1(X)$ which is a subring of $F(X)$. It is shown that the sum of two $z$-ideals in $B_1(X)$ is a $z$-ideal. It is shown that an ideal $I$ in $B_1(X)$ is a $z$-ideal if and only if $\sqrt{I}$ is a $z$-ideal. For any $f\in F(X)$, $f^{-1}(0)$ is denoted by $Z(f)$ and it is called a zero-set.
If $A(X)$ is a subring of $F(X)$, $\emptyset \neq B \subseteq A(X)$ and $S=\bigcup_{b\in B}(X\setminus Z(b))$, then it is shown that there exists a ring homomorphism $\phi:A(X) \to A(S)$ such that ker $\phi = Ann(B)$. An ideal $I$ in $A(X)$ is called a pseudofixed ideal if $\{ cl_X Z(f) | f\in I \}$ has nonempty intersection. Some characterizations of pseudofixed ideals in some subrings of $F(X)$ are given. As a consequence, it is shown that if $X$ is connected, $I$ is a proper free ideal in $C(X)$ and $p\in X$, then there exists nonzero ideal $J$ contained in $I$ such that $p\in \cap Z[J]$. An arbitrary subring $A(X)$ of $F(X)$ such that every $f\in A(X)$ with $Z(f)=\emptyset$ is unit are studied. A subring $A(X)$ of $F(X)$ such that $g\circ f \in A(X)$, where $g\in C(\mathbb{R})$ and $f\in A(X)$ are investigated. A contravariant fanctor from category of topological spaces and continuous maps between them to category of commutative rings with ring homomorphism is investigated.

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References

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Mathematics and Society
Volume 6, Issue 4 - Serial Number 4
December 2021
Pages 29-47

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History

  • Receive Date: 26 March 2022
  • Revise Date: 30 June 2022
  • Accept Date: 30 July 2022
  • Publish Date: 20 February 2022

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