Published in Algebra universalis
The main purpose of this paper is a wide generalization of one of the results abstract algebraic geometry begins with, namely of the fact that the prime spectrum Spec(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \se...
Published in Algebra universalis
We abstract the notion of fraction-density of f-rings (introduced by Anthony Hager and Jorge Martínez) to algebraic frames. We say an algebraic frame with the finite intersection property on compact elements is fraction-dense if each of its polars is a polar of a compact element. This turns out to be a “conservative” extension of the fraction-densi...
Published in Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
In this paper, using the canonical correspondence between the idempotents and clopens, we obtain several new results on lifting idempotents. The Zariski clopens of the maximal spectrum are precisely determined, then as an application, lifting idempotents modulo the Jacobson radical is characterized. Lifting idempotents modulo an arbitrary ideal is ...
Published in Research in the Mathematical Sciences
In 1971, Lipman (Am J Math 93:649–685, 1971) introduced the notion of strict closure of a ring in another, and established the underlying theory in connection with a conjecture of O. Zariski. In this paper, for further developments of the theory, we investigate three different topics related to strict closure of rings. The first one concerns constr...
Published in Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
In this paper, new advances on the understanding the structure of p.p. rings and their generalizations have been made. Especially among them, it is proved that a commutative ring R is a generalized p.p. ring if and only if R is a generalized p.f. ring and its minimal spectrum is Zariski compact, or equivalently, R/N\documentclass[12pt]{minimal} \us...
Published in Communications in Algebra
A commutative ring R is stable if every non-zero ideal I of R is projective over its ring of endomorphisms. Motivated by a paper of Bass in the 1960s, stable rings have received wide attention in the literature ever since then. Much is known on the algebraic structure of stable rings and on the relationship of stability with other algebraic propert...
Published in Bollettino Della Unione Matematica Italiana (2008)
Every Krull monoid has a transfer homomorphism onto a monoid of zero-sum sequences over a subset of its class group. This transfer homomorphism is a crucial tool for studying the arithmetic of Krull monoids. In the present paper, we strengthen and refine this tool for Krull monoids with finitely generated class group.
Published in Discussiones Mathematicae - General Algebra and Applications
Let R be a finite commutative ring with identity. The idempotent graph of R is the simple undirected graph I(R) with vertex set, the set of all nontrivial idempotents of R and two distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we have determined all isomorphism classes of finite commutative rings with identity whose I(...
Published in Algebra universalis
In this paper, we study principal element lattices, Prüfer lattices and Q-lattices. Also, a necessary and sufficient condition is given for a principally generated C-lattice to be a finite direct product of proper Dedekind domains.
Published in Open Mathematics
Let ( Γ , ≤ ) ({\mathrm{\Gamma}},\le ) be a strictly ordered monoid, and let Γ ⁎ = Γ \ { 0 } {{\mathrm{\Gamma}}}^{\ast }\left={\mathrm{\Gamma}}\backslash \{0\} . Let D ⊆ E D\subseteq E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D. Set D + 〚 E Γ ⁎ , ≤ 〛 ≔ f ∈ 〚 E Γ , ≤ 〛 | f ( 0 ) ∈ D and D + 〚 I Γ ⁎ ,...
