Let $ R $ be a commutative noetherian ring. In this paper, we study specialization-closed
subsets of $\mathrm {Spec} R $. More precisely, we first characterize the specialization …

All rings considered in this paper are commutative with unity. A ring R is called a PF-ring if
every principal ideal a R is a flat R-module, and it is called a PP-ring if every principal ideal a …

We show that π-regular rings and clean rings can be completely characterized by
topological properties of their prime spectrums respectively. In addition, we give some …

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) Spec …

We prove that neat and coneat submodules of a module coincide when is a commutative
ring such that every maximal ideal is principal, extending a recent result by Fuchs. We …

THEOREM 1. Assumptions.(i) N is finitely presented as A-left-module, EndA (N) finitely
generated as R-module and M a direct summand in a direct sum of finitely presented A …

A right rnodule M over a ring R is said to have a unimodular element (UME) if there exists ue
M such that uR is a direct summand of M canonically isomorphic to R., Thus, M has a UME iff …

We Show that:(a) closures of a constructible subsets of real spectrum (Spec r A) of complete
noetherian local ring A with formally real residue field R are constructible.(b) The connected …

0. Introduction. Let<€ be the category of commutative rings with unit, and regard Spec (as in
[1]) as a contra variant functor from# to 3~, the category of topological spaces and …

Stable homotopy theory studies the homotopy category of spectra. This category has a
symmetric monoidal smash product which allows the definition of ring spectra 'up to …