Let $L$ be a completely regular frame and $mathcal{R}L$ be the ‎ring of continuous real-valued functions on $L$‎. ‎We show that the‎ ‎lattice $Zid(mathcal{R}L)$ of $z$-ideals of $mathcal{R}L$ is a‎ ‎normal coherent Yosida frame‎, ‎which extends the corresponding $C(X)$‎ ‎result of Mart'{i}nez and Zenk‎. ‎This we do by exhibiting‎ ‎$Zid(mathcal{R}L)$ as a quotient of $Rad(mathcal{R}L)$‎, ‎the‎ ‎...

let $l$ be a completely regular frame and $mathcal{r}l$ be the ‎ring of continuous real-valued functions on $l$‎. ‎we show that the‎ ‎lattice $zid(mathcal{r}l)$ of $z$-ideals of $mathcal{r}l$ is a‎ ‎normal coherent yosida frame‎, ‎which extends the corresponding $c(x)$‎ ‎result of mart'{i}nez and zenk‎. ‎this we do by exhibiting‎ ‎$zid(mathcal{r}l)$ as a quotient of $rad(mathcal{r}l)$‎, ‎the‎ ‎...

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 ...

In this paper, we study a generalization of z-ideals in the ring C(X) of continuous real valued functions on a completely regular Hausdorff space X. The notion of a weak ideal and naturally a weak z-ideal and a prime weak ideal are introduced and it turns out that they behave such as z-ideals in C(X).

in this paper a particular case of z-ideals, called strongly z-ideal, is defined by introducing zero sets in pointfree topology. we study strongly z-ideals, their relation with z-ideals and the role of spatiality in this relation. for strongly z-ideals, we analyze prime ideals using the concept of zero sets. moreover, it is proven that the intersection of all zero sets of a prime ideal of c(l),...

in this paper, we introduce a method by which we can find a close connection between the set of prime $z$-ideals of $c(x)$ and the same of $c(y)$, for some special subset $y$ of $x$. for instance, if $y=coz(f)$ for some $fin c(x)$, then there exists a one-to-one correspondence between the set of prime $z$-ideals of $c(y)$ and the set of prime $z$-ideals of $c(x)$ not containing $f$. moreover, c...

In this paper a particular case of z-ideals, called strongly z-ideal, is defined by introducing zero sets in pointfree topology. We study strongly z-ideals, their relation with z-ideals and the role of spatiality in this relation. For strongly z-ideals, we analyze prime ideals using the concept of zero sets. Moreover, it is proven that the intersection of all zero sets of a prime ideal of C(L),...

Let P := k [X,Y, Z] be a polynomial ring over an algebraic closed field k and (X, Y , Z, XY Z, XY Z ) · k [X,Y, Z] an (X,Y, Z)-primary ideal in P , (m, n, l, a, b, c, d, e, f are integers). The ideal Q=(X, Y , Z, XY Z, XY Z ) ·R is (X,Y, Z) · R-primary ideal in the local ring R = k [X,Y, Z](x,y,z). In this short note we give a formula for the calculation of Samuel multiplicity e0(Q,R) of the id...