The minimal prime decomposition for semiprime ideals is defined and studied on z-ideals of C(X). The necessary and sufficient condition for existence of the minimal prime decomposition of a z-ideal / is given, when / satisfies one of the following conditions: (i) / is an intersection of maximal ideals. (ii) I is an intersection of O , s, when X is basically disconnected. (iii) I=O , when x X h...

Assumed that (S,+,.) is a semiring. Semiring algebra structure as generalization of ring. A set I⊆S called an ideal over semiring S if for any α,β∈I, we have α-β∈I and sα=αs∈I every s in S. Based on this definition, there special condition namely prime P, when αβ∈P, then could prove α or β are elements P. Furthermore, I irreducible Ia intersection from B S, I=A I=B. We also know the strongly no...

‎In this paper‎, ‎we define and study the notion of zero elements in topoframes; a topoframe is a pair‎ ‎$(L‎, ‎tau)$‎, ‎abbreviated $L_{ tau}$‎, ‎consisting of a frame $L$ and a‎ ‎subframe $ tau $ all of whose elements are complemented elements in‎ ‎$L$‎. ‎We show that‎ ‎the $f$-ring $ mathcal{R}(L_tau)$‎, ‎the set of $tau$-real continuous functions on $L$‎, ‎is uniformly complete‎. ‎Also‎, ‎t...

In this paper we explore the interrelationship between the theory of polynomial ideals and certain branches of analysis including multivariate approximation theory and linear partial differential equations. Specifically, we use the perturbation technique from analysis to study the codimension of a multivariate polynomial ideal. Let K be a field. In this paper, K will often be the field C of com...

We say that a Tychonoff space X has computable z-radicals if for all ideals a of C(X), the smallest z-ideal containing a is generated as an ideal by all the s◦f , where f is in a and s is a continuous function IR −→ IR with s−1(0) = {0}. We show that every cozero set of a compact space has computable z-radicals and that a subset X of IR has computable z-radicals if and only if X is locally closed.

We say that a Tychonoff space X has computable z-radicals if for all ideals a of C(X), the smallest z-ideal containing a is generated as an ideal by all the s ◦ f , where f is in a and s is a continuous function IR −→ IR with s−1(0) = {0}. We show that every cozero set of a compact space has computable z-radicals and that a subset X of IR has computable z-radicals if and only if X is locally cl...

b = 1. The index set I must be of the form I = A [ B [ C where A = f1g B = fz + 1 j z 2 C0; z + 1 2 C0g and C = f0z j z 2 C1; z + 1 2 C1g: Observe that these three sets are pairwise-disjoint. Therefore, The other cases can be treated similarly. ACKNOWLEDGMENT The authors wish to thank the anonymous referee for careful review of the original manuscript and helpful comments. Double-check of the c...