H-local Domains Research Articles

When C(X) is an h-local ring

An h-local domain is a domain for which each nonzero prime ideal is contained in a unique maximal ideal and each nonzero element has finite character. In his dissertation, A. Omairi generalizes the notion of an h-local domain to rings with zero-divisors by restricting the definition to regular ideals. In this article, we give a characterization of when C(X) the ring of continuous real-valued functions on a space X is an h-local ring. We are left with more questions than answers.

∗-Almost super-homogeneous ideals in ∗-h-local domains

In this paper, we introduce ∗-almost independent rings of Krull type (∗-almost IRKTs) and ∗-almost generalized Krull domains (∗-almost GKDs) in the general theory of almost factoriality, neither of which need be integrally closed. This fills a gap left in [D. D. Anderson and M. Zafrullah, On∗-Semi-Homogeneous Integral Domains, Advances in Commutative Algebra (Springer, Singapore, 2019)]. We characterize them by ∗-almost super-SH domains, where a domain [Formula: see text] is called a ∗-almost super-SH domain if every nonzero proper principal ideal of [Formula: see text] is a ∗-product of ∗-almost super-homogeneous ideals. We prove that (1) a domain [Formula: see text] is a ∗-almost IRKT if and only if [Formula: see text] is a ∗-almost super-SH domain, (2) a domain is a ∗-almost GKD if and only if [Formula: see text] is a type 1 ∗-almost super-SH domain and (3) a domain [Formula: see text] is a ∗-almost IRKT and an AGCD-domain if and only if [Formula: see text] is a ∗-afg-SH domain. Further, we characterize them by their integral closures. For example, we prove that a domain [Formula: see text] is an almost IRKT if and only if [Formula: see text] is a root extension with [Formula: see text] [Formula: see text]-linked under [Formula: see text] and [Formula: see text] is an IRKT. Examples are given to illustrate the new concepts.

On coprimely structured rings

In this paper, we define coprimely structured rings, which are the generalization of strongly 0-dimensional rings. Furthermore, we investigate coprimely structured rings and give some relations between other rings such as Artinian rings, strongly 0-dimensional rings, and h-local domains.

Localizations of endomorphism rings and fixpoints

Let U E ‘8 be an object of an abeiian category with exact direct limits. Let A = [U, U] be the endomorphism ring of U and let Modn be the category of right A-modules. For each X E 2l the ring A acts on the group [U, X] of all morphisms U X by means of composition and the functor [U, -I: % + Mod,r , X [U, A] has a left adjoint @)A U: Mod, + %. Let FI the motivation being the theorem of Gabriel and Popescu [4]. For this, consider the filter 5 of all right ideals I C .A which cover U in the sense that U = uvel im y, where im y denotes the image of y: U + U. Let (Mod& be the full subcategory of all modules YE ModA such that the restriction map [A, Y] [I, Y] is bijective for each 1~ 3. Then by [S, 8.6(b)] the inclusion (Mod& -+ Mod=,, has a left adjoint 3-10~: Mod,, -+ (Mod& (= g-localization). In general the functor @lot is not exact. Roughly speaking, the main result is a necessary and sufficient condition on C for the composite F,(S) (Mod& , X$‘$loc[tr, X] to be an equivalence and for I in particular when % is a category of modules over an h-local domain or a Dedekind domain. If U is a direct sum of finitely generated modules or if U is divisible or torsion or algebraically

Decompositions of modules and matrices

A canonical form for a module M over a commutative ring R is a decomposition M £ R/^ © • • • 0 #//„, where the Ij are ideals oiR and I x £ ••• £ /„.A complete structure theory is developed for those rings for which every finitely generated module has a canonical form. The (possibly larger) class of rings, for which every finitely generated module is a direct sum of cyclics, is also considered, and partial results are obtained for rings with fewer than 2 prime ideals. For example, if R is countable and every finitely generated R-module is a direct sum of cyclics, then R is a principal ideal ring. Finally, some topological criteria are given for Hermite rings and elementary divisor rings. All rings in this announcement are commutative with 1, and all modules are unital. A canonical form for an K-module M is a decomposition M ^ Rjlx © • • • © R/In, where Ix g • • • g In # jR.IfM has a canonical form, the ideals Ij are uniquely determined [K]. A CF-ring is a ring for which every finitely generated direct sum of cyclics has a canonical form. It can be shown that R is CF if and only if R/I © R/J S R/(I nJ)® R/(I + J) for every pair of ideals /, J. By a valuation ring we shall mean a ring, possibly with zero-divisors, whose lattice of ideals is totally ordered. A ring R is arithmetical, provided the local ring Rm is a valuation ring for each maximal ideal m. Finally, an Zi-local domain [ M l ] is an integral domain such that (1) every nonzero ideal is contained in only finitely many maximal ideals, and (2) every nonzero prime ideal is contained in a unique maximal ideal. THEOREM 1. Every CF-ring is a finite direct product of indecomposable C F-rings. The indecomposable CF-ring s are precisely the rings R such that (i) R is arithmetical, (ii) R has a unique minimal prime P, (iii) R/P is an h-local domain, and (iv) every ideal contained in P is comparable with every ideal of R. Thus valuation rings and arithmetical /z-local domains are CF-rings. AMS (MOS) subject classifications (1970). Primary 13C05, 13F05; Secondary 15A21.