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- W2994309114 abstract "In this article we observe that $C(X)$ is integral over $I+mathbb R$ if and only if $I$ is of the form of a finite intersection of real maximal ideals. The integral closure of a subring $I+mathbb R$ is investigated and it turns out that $I+mathbb R$ is integrally closed if and only if $I$ is semiprime and $theta (I)$ is connected. We show that $I^u+mathbb R$, where $I^u$ is the uniform closure of the ideal $I$, is always a ring of quotients of $I+mathbb R$ and $C(X)$ is a ring of quotients of $I+mathbb R$ if and only if $I$ is either essential (large) or a maximal ideal generated by an idempotent. Maximal subrings of the form $I+mathbb R$ of $C(X)$ are characterized and it is shown that $[M^p]^u+mathbb R$ (which is isomorphic to $C(Xcup {p})$, where $Xcup {p}$ is a subspace of $beta X$) is not contained in a proper maximal subring of $C(X)$. We have observed that for each subring of the form $I+mathbb R$ of $C(X)$, the sum of every two prime ideals of $I+mathbb R$ is prime or all of $I+mathbb R$ if and only if $X$ is an $F$-space. Ideals $I$ are characterized for which $I+mathbb R$ is a regular ring and for every $z$-ideal $I$, we have shown that the sum of every two $z$-ideals of $I+mathbb R$ is a $z$-ideal or all of $I+mathbb R$. Finally, some algebraic and topological properties are given for which $I+mathbb R$ is a self-injective ring or a uem-ring." @default.
- W2994309114 created "2019-12-13" @default.
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- W2994309114 date "2019-08-01" @default.
- W2994309114 modified "2023-10-13" @default.
- W2994309114 title "On subrings of the form $I+mathbb {R}$ of $C(X)$" @default.
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- W2994309114 doi "https://doi.org/10.1216/jca-2019-11-4-479" @default.
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