On nonregular ideals and z°-ideals in C(X)

Azarpanah, F.; Karavan, M.

doi:10.1007/s10587-005-0030-0

Cited by 17 publications

(19 citation statements)

References 8 publications

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“…We can use the general theory of z-ideals to obtain some of their properties. Some of the statements are proven for z-ideals in C (X) in [3], [22]. Proposition 5.3.…”

Section: If I ⊳ K Is Pseudoprime Imentioning

confidence: 99%

Ideals in the Ring of Colombeau Generalized Numbers

Vernaeve

2010

Communications in Algebra

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In this paper, the structure of the ideals in the ring of Colombeau generalized numbers is investigated. Connections with the theories of exchange rings, Gelfand rings and lattice-ordered rings are given. Characterizations for prime, projective, pure and topologically closed ideals are given, answering in particular the questions about prime ideals in [1]. Also z-ideals in the sense of [21] are characterized. The quotient rings modulo maximal ideals are shown to be canonically isomorphic with nonstandard fields of asymptotic numbers. Finally, a detailed study of the ideals allows us to prove that (under some set-theoretic assumption) the Hahn-Banach extension property does not hold for a large class of topological modules over the ring of Colombeau generalized numbers.

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“…We can use the general theory of z-ideals to obtain some of their properties. Some of the statements are proven for z-ideals in C (X) in [3], [22]. Proposition 5.3.…”

Section: If I ⊳ K Is Pseudoprime Imentioning

confidence: 99%

Ideals in the Ring of Colombeau Generalized Numbers

Vernaeve

2010

Communications in Algebra

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“…Conversely, it suffices to show that every prime z -ideal consisting entirely of zerodivisors is a z • -ideal, by [[ [9]], Theorem 4.2]. To this end, we just notice that every prime ideal consisting entirely of zerodivisors is an r -ideal.…”

Section: Z(f ) ⊆ Z(gh)mentioning

confidence: 99%

“…For more information about the ideals in C(X), see [ [7], [10], [12], [16]], and for details about topological spaces, see [[14], [16]]. …”

Section: Introductionmentioning

confidence: 99%

$r$-ideals in commutative rings

Mohamadian¹

2015

Turk J Math

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Turkish Journal of Mathematics h t t p : / / j o u r n a l s . t u b i t a k . g o v . t r / m a t hAbstract: In this article we introduce the concept of r -ideals in commutative rings (note: an ideal I of a ring R is called r -ideal, if ab ∈ I and Ann(a) = (0) imply that b ∈ I for each a, b ∈ R ). We study and investigate the behavior of r -ideals and compare them with other classical ideals, such as prime and maximal ideals. We also show that some known ideals such as z • -ideals are r -ideals. It is observed that if I is an r -ideal, then so too is a minimal prime ideal of I . We naturally extend the celebrated results such as Cohen's theorem for prime ideals and the Prime AvoidanceLemma to r -ideals. Consequently, we obtain interesting new facts related to the Prime Avoidance Lemma. It is also shown that R satisfies property A (note: a ring R satisfies property A if each finitely generated ideal consisting entirely of zerodivisors has a nonzero annihilator) if and only if for every r -ideal. Using this concept in the context of C(X) , we show that every r -ideal is a z • -ideal if and only if X is a ∂ -space (a space in which the boundary of any zeroset is contained in a zeroset with empty interior). Finally, we observe that, although the socle of C(X) is never a prime ideal in C(X) , the socle of any reduced ring is always an r -ideal.

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“…We first obtain a characterisation for this phenomenon for reduced f -rings. It will generalise the equivalence of conditions (i) and (ii) in [1,Theorem 3.2]. We start with a lemma which is itself an f -ring version of [1, Lemma 3.1].…”

Section: When Prime D-ideals Are Minimal or Maximalmentioning

confidence: 99%

When certain prime ideals in rings of continuous functions are minimal or maximal

Dube

Nsayi

2015

Topology and its Applications

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A Tychonoff space X is called a quasi m-space if every prime d-ideal of the ring C(X) is either a maximal ideal or a minimal prime ideal. These spaces were characterised by Azarpanah and Karavan. In this paper we look at some properties of these spaces from a ring-theoretic perspective. We observe, for instance, that among subspaces which inherit this property are (i) cozero subspaces, (ii) dense z-embedded subspaces, and (iii) regular-closed subspaces among the normal quasi m-spaces. The ring-theoretic approach actually yields the above results within the broader context of frames. The latter part of the paper discusses completely regular frames L for which every prime z-ideal in the ring RL is a maximal ideal or a minimal prime ideal.

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On nonregular ideals and z°-ideals in C(X)

Cited by 17 publications

References 8 publications

Ideals in the Ring of Colombeau Generalized Numbers

Ideals in the Ring of Colombeau Generalized Numbers

$r$-ideals in commutative rings

When certain prime ideals in rings of continuous functions are minimal or maximal

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