It is well-known that the sum of two $z$-ideals in $C(X)$ is either $C(X)$ or a $z$-ideal.

The main aim of this paper is to study the sum of strongly $z$-ideals in ${mathcal{R}} L$, the ring of real-valued continuous functions on a frame $L$.

For every ideal $I$ in ${mathcal{R}} L$, we introduce the biggest strongly $z$-ideal included in $I$ and the smallest strongly $z$-ideal containing $I$,

denoted by $I^{sz}$ and $I_{sz}$, respectively.

We study some properties of $I^{sz}$ and $I_{sz}$.

Also, it is observed that the sum of any family of minimal prime ideals in the ring ${mathcal{R}} L$ is either ${mathcal{R}} L$ or a prime strongly $z$-ideal in ${mathcal{R}} L$.

In particular, we show that the sum of two prime ideals in ${mathcal{R}} L$ such that are not a chain, is a prime strongly $z$-ideal.

Type of Study: Research paper |
Subject:
Special

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