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...

The factorization of ζo(s) is the main issue. After giving a definition of this zeta function, we will see that the factorization is equivalent to understanding the behavior of rational primes in the extension ring Z[ω] of Z: do they stay prime, or do they factor as products of primes in Z[ω]? A complication is that the rings Z[ω] are rarely principal ideal domains. To delay contemplation of th...

Theorem ([11, Theorem 2, page 282]). Let R be a prime ring, L a noncommutative Lie ideal of R and d 6= 0 a derivation of R. If [d(x), x] ∈ Z(R), for all x ∈ L, then either R is commutative, or char(R) = 2 and R satisfies s4, the standard identity in 4 variables. Here we will examine what happens in case [d(x), x]n ∈ Z(R), for any x ∈ L, a noncommutative Lie ideal of R and n ≥ 1 a fixed integer....

A frame $L$ is called {it coz-dense} if $Sigma_{coz(alpha)}=emptyset$ implies $alpha=mathbf 0$. Let $mathcal RL$ be the ring of real-valued continuous functions on a coz-dense and completely regular frame $L$. We present a description of the socle of the ring $mathcal RL$ based on minimal ideals of $mathcal RL$ and zero sets in pointfree topology. We show that socle of $mathcal RL$ is an essent...

Let X be a nonsingular algebraic variety. Suppose Z ⊆ X is a closed subscheme of X, with ideal sheaf IZ . When Z has codimension one in X, everything is as nice as it could be: IZ is a locally free sheaf, in fact a line bundle, and Z can locally be defined by a single equation. But starting in codimension two, all these pleasant things are usually false. To begin with, not every closed subschem...

For any abelian Polish σ-compact group H there exist an Fσ ideal Z ⊆ P (N) and a Borel Z -approximate homomorphism f : H → HN which is not Z -approximable by a continuous true homomorphism g : H → HN .

In this paper, we draw connections between ideal lattices and multivariate polynomial rings over integers using Gröbner bases. Univariate ideal lattices are ideals in the residue class ring, Z[x]/〈f〉 (here f is a monic polynomial) and cryptographic primitives have been built based on these objects. Ideal lattices in the univariate case are generalizations of cyclic lattices. We introduce the no...

Let Z be a finite set of double points in P 1 × P 1 and suppose further that X, the support of Z, is arithmetically Cohen-Macaulay (ACM). The scheme Z is rarely ACM. In this note we present an algorithm, which depends only upon a combinatorial description of X, for the bigraded Betti numbers of I Z , the defining ideal of Z.