scholarly journals On feckly clean rings

2015 ◽  
Vol 14 (04) ◽  
pp. 1550046 ◽  
Author(s):  
Huanyin Chen ◽  
H. Kose ◽  
Y. Kurtulmaz
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Clean Rings ◽  
Maximal Ideals ◽  
Unique Maximal Ideal

A ring R is feckly clean provided that for any a ∈ R there exists an element e ∈ R and a full element u ∈ R such that a = e + u, eR(1 - e) ⊆ J(R). We prove that a ring R is feckly clean if and only if for any a ∈ R, there exists an element e ∈ R such that V(a) ⊆ V(e), V(1 - a) ⊆ V(1 - e) and eR(1 - e) ⊆ J(R), if and only if for any distinct maximal ideals M and N, there exists an element e ∈ R such that e ∈ M, 1 - e ∈ N and eR(1 - e) ⊆ J(R), if and only if J- spec (R) is strongly zero-dimensional, if and only if Max (R) is strongly zero-dimensional and every prime ideal containing J(R) is contained in a unique maximal ideal. More explicit characterizations are also discussed for commutative feckly clean rings.

scholarly journals On 2-absorbing ideals of commutative rings

2007 ◽  
Vol 75 (3) ◽  
pp. 417-429 ◽  
Author(s):  
Ayman Badawi
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Integral Domain ◽  
Commutative Rings ◽  
Dedekind Domain ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Maximal Ideals ◽  
Noetherian Domain ◽  
Valuation Domains

Suppose that R is a commutative ring with 1 ≠ 0. In this paper, we introduce the concept of 2-absorbing ideal which is a generalisation of prime ideal. A nonzero proper ideal I of R is called a 2-absorbing ideal of R if whenever a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I. It is shown that a nonzero proper ideal I of R is a 2-absorbing ideal if and only if whenever I1I2I3 ⊆ I for some ideals I1,I2,I3 of R, then I1I2 ⊆ I or I2I3 ⊆ I or I1I3 ⊆ I. It is shown that if I is a 2-absorbing ideal of R, then either Rad(I) is a prime ideal of R or Rad(I) = P1 ⋂ P2 where P1,P2 are the only distinct prime ideals of R that are minimal over I. Rings with the property that every nonzero proper ideal is a 2-absorbing ideal are characterised. All 2-absorbing ideals of valuation domains and Prüfer domains are completely described. It is shown that a Noetherian domain R is a Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R. If RM is Noetherian for each maximal ideal M of R, then it is shown that an integral domain R is an almost Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R.

scholarly journals On 2-absorbing ideals of commutative semirings

2019 ◽  
Vol 19 (02) ◽  
pp. 2050034
Author(s):  
H. Behzadipour ◽  
P. Nasehpour
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Ideal Formula ◽  
Minimal Prime ◽  
Unique Maximal Ideal

In this paper, we investigate 2-absorbing ideals of commutative semirings and prove that if [Formula: see text] is a nonzero proper ideal of a subtractive valuation semiring [Formula: see text] then [Formula: see text] is a 2-absorbing ideal of [Formula: see text] if and only if [Formula: see text] or [Formula: see text] where [Formula: see text] is a prime ideal of [Formula: see text]. We also show that each 2-absorbing ideal of a subtractive semiring [Formula: see text] is prime if and only if the prime ideals of [Formula: see text] are comparable and if [Formula: see text] is a minimal prime over a 2-absorbing ideal [Formula: see text], then [Formula: see text], where [Formula: see text] is the unique maximal ideal of [Formula: see text].

scholarly journals MAXIMAL IDEALS IN THE ALGEBRA OF OPERATORS ON CERTAIN BANACH SPACES

2002 ◽  
Vol 45 (3) ◽  
pp. 523-546 ◽  
Author(s):  
Niels Jakob Laustsen
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Maximal Ideal ◽  
Operator Ideal ◽  
Linear Operators ◽  
Weakly Compact ◽  
Maximal Ideals ◽  
Hereditarily Indecomposable Banach Space ◽  
Unique Maximal Ideal ◽  
The Ideal

AbstractFor a Banach space $\mathfrak{X}$, let $\mathcal{B}(\mathfrak{X})$ denote the Banach algebra of all continuous linear operators on $\mathfrak{X}$. First, we study the lattice of closed ideals in $\mathcal{B}(\mathfrak{J}_p)$, where $1 \lt p \t \infty$ and $\mathfrak{J}_p$ is the $p$th James space. Our main result is that the ideal of weakly compact operators is the unique maximal ideal in $\mathcal{B}(\mathfrak{J}_p)$. Applications of this result include the following.(i) The Brown–McCoy radical of $\mathcal{B}(\mathfrak{X})$, which by definition is the intersection of all maximal ideals in $\mathcal{B}(\mathfrak{X})$, cannot be turned into an operator ideal. This implies that there is no ‘Brown–McCoy’ analogue of Pietsch’s construction of the operator ideal of inessential operators from the Jacobson radical of $\mathcal{B}(\mathfrak{X})/\mathcal{A}(\mathfrak{X})$.(ii) For each natural number $n$ and each $n$-tuple $(m_1,\dots,m_n)$ in $\{k^2\mid k\in\mathbb{N}\}\cup\{\infty\}$, there is a Banach space $\mathfrak{X}$ such that $\mathcal{B}(\mathfrak{X})$ has exactly $n$ maximal ideals, and these maximal ideals have codimensions $m_1,\dots,m_n$ in $\mathcal{B}(\mathfrak{X})$, respectively; the Banach space $\mathfrak{X}$ is a finite direct sum of James spaces and $\ell_p$-spaces.Second, building on the work of Gowers and Maurey, we obtain further examples of Banach spaces $\mathfrak{X}$ such that all the maximal ideals in $\mathcal{B}(\mathfrak{X})$ can be classified. We show that the ideal of strictly singular operators is the unique maximal ideal in $\mathcal{B}(\mathfrak{X})$ for each hereditarily indecomposable Banach space $\mathfrak{X}$, and we prove that there are $2^{2^{\aleph_0}}$ distinct maximal ideals in $\mathcal{B}(\mathfrak{G})$, where $\mathfrak{G}$ is the Banach space constructed by Gowers to solve Banach’s hyperplane problem.AMS 2000 Mathematics subject classification: Primary 47D30; 47D50; 46H10; 16D30

scholarly journals Intuitionistic fuzzy normed prime and maximal ideals

2021 ◽  
Vol 6 (10) ◽  
pp. 10565-10580
Author(s):  
Nour Abed Alhaleem ◽  
◽  
Abd Ghafur Ahmad
Keyword(s):  
Characteristic Function ◽  
Prime Ideal ◽  
Maximal Ideal ◽  
Prime Ideals ◽  
Intuitionistic Fuzzy ◽  
Normed Ideal ◽  
Maximal Ideals

<abstract><p>Motivated by the new notion of intuitionistic fuzzy normed ideal, we present and investigate some associated properties of intuitionistic fuzzy normed ideals. We describe the intrinsic product of any two intuitionistic fuzzy normed subsets and show that the intrinsic product of intuitionistic fuzzy normed ideals is a subset of the intersection of these ideals. We specify the notions of intuitionistic fuzzy normed prime ideal and intuitionistic fuzzy normed maximal ideal, we present the conditions under which a given intuitionistic fuzzy normed ideal is considered to be an intuitionistic fuzzy normed prime (maximal) ideal. In addition, the relation between the intuitionistic characteristic function and prime and maximal ideals is generalized. Finally, we characterize relevant properties of intuitionistic fuzzy normed prime ideals and intuitionistic fuzzy normed maximal ideals.</p></abstract>

Projective Ideals of Finite Type

1969 ◽  
Vol 21 ◽  
pp. 1057-1061 ◽  
Author(s):  
William W. Smith
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Finite Type ◽  
Local Ring ◽  
Maximal Ideal ◽  
Quotient Ring ◽  
Finitely Generated ◽  
Unique Maximal Ideal

The main results in this paper relate the concepts of flatness and projectiveness for finitely generated ideals in a commutative ring with unity. In this discussion the idea of a multiplicative ideal is used.Definition.An ideal Jis multiplicative if and only if whenever I is an ideal with I ⊂ J there exists an ideal Csuch that I = JC.Throughout this paper Rwill denote a commutative ring with unity. If I and Jare ideals of R,then I: J = {x| xJ ⊂ I}. By “prime ideal” we will mean “proper prime ideal” and Specie will denote this set of ideals. Ris called a local ring if it has a unique maximal ideal (the ring need not be Noetherian). If P is in Spec R then RPis the quotient ring formed using the complement of P.

scholarly journals NONCOMMUTATIVE HILBERT RINGS

2004 ◽  
Vol 03 (04) ◽  
pp. 437-443 ◽  
Author(s):  
ALGIRDAS KAUCIKAS ◽  
ROBERT WISBAUER
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Commutative Rings ◽  
The Other ◽  
Prime Ideals ◽  
Natural Form ◽  
Noncommutative Rings ◽  
Ideal Formula ◽  
Contraction Property ◽  
Maximal Ideals

Commutative rings in which every prime ideal is the intersection of maximal ideals are called Hilbert (or Jacobson) rings. This notion was extended to noncommutative rings in two different ways by the requirement that prime ideals are the intersection of maximal or of maximal left ideals, respectively. Here we propose to define noncommutative Hilbert rings by the property that strongly prime ideals are the intersection of maximal ideals. Unlike for the other definitions, these rings can be characterized by a contraction property: R is a Hilbert ring if and only if for all n∈ℕ every maximal ideal [Formula: see text] contracts to a maximal ideal of R. This definition is also equivalent to [Formula: see text] being finitely generated as an [Formula: see text]-module, i.e., a liberal extension. This gives a natural form of a noncommutative Hilbert's Nullstellensatz. The class of Hilbert rings is closed under finite polynomial extensions and under integral extensions.

Maximal Ideals in Some Spaces of Bounded Linear Operators

2018 ◽  
Vol 61 (1) ◽  
pp. 251-264
Author(s):  
Denny H. Leung
Keyword(s):  
Banach Spaces ◽  
Maximal Ideal ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Direct Sum ◽  
Maximal Ideals ◽  
Unique Maximal Ideal

We add to the list of Banach spaces X for which it is known that the space of bounded linear operators on X has a unique maximal ideal. In particular, the result holds if X is a subsymmetric direct sum of ℓp or of the Schlumprecht space S. We also show that two recently identified ideals in L(Jp), where Jp is the pth James space, each contains a unique maximal ideal.

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

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