We extend the basic fact that every ideal of a finite dimensional semisimple Lie algebra has a unique complement to the case of closed ideals of prosemisimple Lie algebras. We prove that if A is a closed ideal of a prosemisimple Lie algebra L = lim ←−−Ln (n ∈ N), where the Ln are finite dimensional semisimple Lie algebras, then there exists a unique ideal B of L such that L = A ⊕ B. Mathematics...

This paper studies the Ratliff-Rush closure and the integral closure of an ideal in a valuation and Prüfer domains. By definition, the RatliffRush closure (respectively the integral closure) of an ideal I of a domain R is the ideal Given by Ĩ := S (I :R I ) (respectively I := {x ∈ R|x satisfies an equation of the form x + a1x + · · ·+ ak = 0, where ai ∈ I i for each i ∈ {1, . . . , k}}). An ide...

This paper studies the Ratliff-Rush closure and the integral closure of an ideal in a valuation and Prüfer domains. By definition, the RatliffRush closure (respectively the integral closure) of an ideal I of a domain R is the ideal given by Ĩ := S (I :R I ) (respectively I := {x ∈ R|x satisfies an equation of the form x + a1x + · · ·+ ak = 0, where ai ∈ I i for each i ∈ {1, . . . , k}}). An ide...

In this paper, we define the new notion of quasi-prime ideal which generalizes at once both prime ideal and primary ideal notions. Then a natural topology on the set of quasi-prime ideals of a ring is introduced which admits the Zariski topology as a subspace topology. The basic properties of the quasi-prime spectrum are studied and several interesting results are obtained. Specially, it is pro...

For Banach algebras A in a class which includes all group and function algebras, we show that the family of ideals of A with the same hull is typically quite large, containing ascending and descending chains of arbitrary length through any ideal in the family, and that typically a closed ideal of A whose hull meets the Silov boundary of A cannot be countably generated algebraically. Guided by t...

Let us assume we are given a linearly uncountable, holomorphic ideal acting almost everywhere on an ultra-arithmetic graph rK. Recent developments in rational PDE [10] have raised the question of whether D < −1. We show that ξ ≥ F . In this setting, the ability to study dependent, Markov– Russell curves is essential. Recent interest in systems has centered on computing covariant, partially nonn...

In this paper, we define and investigate the notions of πgI-closed sets and πgI-open sets in ideal topological spaces. Then, we define ∨π-sets and ∧π-sets and discuss the relation between them. Also, we give characterizations of πgI-closed sets and πgs-closed sets. A separation axiom stronger than πTI -space is defined and various characterizations are given.

We extend a result from 5], namely that locally nitely generated ideals in the ring R 0 are closed, to show that a (non-homogeneous) ideal I in R 0 , whose associated homogeneous ideal gr T (I) is locally nitely generated, is closed. Furthermore, we prove analogous results to some of the \approximation results" of 4], for non-homogeneous ideals.

The aim of this article is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. In order to provide these soft algebraic structures, the notion of union-soft sets is introduced, and its application to BCK/BCI-algebras is considered. The notions of union-soft algebras, union-soft (commutative) ideals and closed union-soft ideals are int...

Fuzzy ideal set and fuzzy local function of fuzzy ideals with fuzzy topologies were introduced and studied by D. Sarker [10]. The purpose of this paper deals with new sort of fuzzy ideal and fuzzy local function namely fuzzy ideal and r-fuzzy open local function. Many of its characterizations, properties and connections between it and other corresponding fuzzy notions are studied. Also, we intr...