TY - JOUR
T1 - Sums of Strongly z-Ideals and Prime Ideals in ${mathcal{R}} L$
TT -
JF - IJMSI
JO - IJMSI
VL - 15
IS - 1
UR - http://ijmsi.ir/article-1-1025-en.html
Y1 - 2020
SP - 23
EP - 34
KW - Frame
KW - Ring of real-valued continuous functions
KW - z-Ideal
KW - Strongly z-ideal.
N2 - 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.
M3 10.29252/ijmsi.15.1.23
ER -