It is proved that if R is a valuation domain with maximal ideal P and if RL is countably generated for each prime ideal L, then R R is separable if and only RJ is maximal, where J = ∩n∈NP . When R is a valuation domain satisfying one of the following two conditions: (1) R is almost maximal and its quotient field Q is countably generated (2) R is archimedean Franzen proved in [2] that R is separ...

A ring of continuous functions is a ring of the form C(X), the ring of all continuous real-valued functions on a completely regular Hausdorff space X. With each ideal / of C(X), we associate certain subalgebras of C(X), and discuss their structure spaces. We give necessary and sufficient conditions for two ideals in rings of continuous functions to have homeomorphic structure spaces. Introducti...

Proof. First note that the zero element of the ring R/I is 0 + I = I and that a + I = I if and only if a ∈ I. Now suppose that I ⊆ R is a prime ideal and consider nonzero cosets a+ I and b+ I in R/I (i.e. consider a 6∈ I and b 6∈ I). Since I is prime this implies that ab 6∈ I, hence ab + I 6= I and we conclude that R/I is an integral domain. Conversely, let R/I be an integral domain and conside...

Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence and thus satisfy stable Harbourne conjecture. In prime characteristic, can take any radical ideal a ring in regular ring, provided its symbolic powers are given saturations with maximal ideal. Although this property is not suitable for reduction to characteristic p, simi...

This paper has two aims. The first is to study ideals of minors of matrices whose entries are among the variables of a polynomial ring. Specifically, we describe matrices whose ideals of minors of a given size are prime. The “generic” case, where all the entries are distinct variables has been studied extensively (cf. [1] and [2] for a thorough account.) While some special cases, such as catale...

Let N ≡ 1 mod 4 be the negative of a prime, K = Q( √ N) and OK its ring of integers. Let D be a prime ideal in OK of prime norm congruent to 3 modulo 4. Under these assumptions, there exists Hecke characters ψD of K with conductor (D) and infinite type (1, 0). Their L-series L(ψD , s) are associated to a CM elliptic curve A(N,D) defined over the Hilbert class field of K. We will prove a Waldspu...

The conjecture that every algebra norm || • || on C(X) is equivalent to the uniform norm arises naturally from a theorem of Kaplansky in 1949 that necessarily 11/11 > \f\x ( ƒ e C(X)): see [9, 10.1]. The seminal paper on the automatic continuity of homomorphisms from C(X) is the 1960 paper of Bade and Curtis [2] in which, for example, it is proved that there is a discontinuous homomorphism from...

Let $G$ be a monoid with identity $e$. In this paper, first we introduce the notions of $G$-graded hyperrings, graded hyperideals and graded hyperfields in the sense of Krasner hyperring $R$. Also, we define the notion of a greded $R$-hypermodules and some examples are presented. Then we investigate graded maximal, graded prime and graded primary hyperideals of a graded hyperring $R$. Finally, ...