## Abstract

Let $X$ be a Tychonoff space and $A\left(X\right)$ be an intermediate subalgebra of $C\left(X\right)$, i.e., ${C}^{\ast}\left(X\right)\subseteq A\left(X\right)\subseteq C\left(X\right)$. We show that such subrings are precisely absolutely convex subalgebras of $C\left(X\right)$. An ideal $I$ in $A\left(X\right)$ is said to be a ${z}_{A}$-ideal if $Z\left(f\right)\subseteq Z\left(g\right)$, $f\in I$ and $g\in A\left(X\right)$ imply that $g\in I$. We observe that the coincidence of ${z}_{A}$-ideals and $z$-ideals of $A\left(X\right)$ is equivalent to the equality $A\left(X\right)=C\left(X\right)$. This shows that every $z$-ideal in $A\left(X\right)$ need not be a ${z}_{A}$-ideal and this is a point which is not considered by D. Rudd in Theorem 4.1 of *Michigan Math. J. ***17 **(1970), 139–141, or by G. Mason in Theorem 3.3 and Proposition 3.5 of *Canad. Math. Bull. ***23**:4 (1980), 437-443. We rectify the induced misconceptions by showing that the sum of $z$-ideals in $A\left(X\right)$ is indeed a $z$-ideal in $A\left(X\right)$. Next, by studying the sum of $z$-ideals in subrings of the form $I+\mathbb{R}$ of $C\left(X\right)$, where $I$ is an ideal in $C\left(X\right)$, we investigate a wide class of examples of subrings of $C\left(X\right)$ in which the sum of $z$-ideals need not be a $z$-ideal. It is observed that, for every ideal $I$ in $C\left(X\right)$, the sum of any two $z$-ideals in $I+\mathbb{R}$ is a $z$-ideal in $I+\mathbb{R}$ or all of $I+\mathbb{R}$ if and only if $X$ is an $F$-space. This result answers a question raised by Azarpanah, Namdari and Olfati in *J.* *Commut. Algebra ***11**:4 (2019), 479–509.

## Citation

Fariborz Azarpanah. Mehdi Parsinia. "On the sum of $z$-ideals in subrings of $C(X)$." J. Commut. Algebra 12 (4) 459 - 466, Winter 2020. https://doi.org/10.1216/jca.2020.12.459

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