Classes of Commutative Clean Rings

Wolf Iberkleid; Warren Wm. McGovern

Displaying similar documents to “Classes of Commutative Clean Rings”

A canonical directly infinite ring

Mario Petrich, Pedro V. Silva (2001)

Czechoslovak Mathematical Journal

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Let $ℕ$ be the set of nonnegative integers and $ℤ$ the ring of integers. Let $ℬ$ be the ring of $N \times N$ matrices over $ℤ$ generated by the following two matrices: one obtained from the identity matrix by shifting the ones one position to the right and the other one position down. This ring plays an important role in the study of directly finite rings. Calculation of invertible and idempotent elements of $ℬ$ yields that the subrings generated by them coincide. This subring is the sum of the ideal $ℱ$ consisting...

When does the Katětov order imply that one ideal extends the other?

Paweł Barbarski, Rafał Filipów, Nikodem Mrożek, Piotr Szuca (2013)

Colloquium Mathematicae

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We consider the Katětov order between ideals of subsets of natural numbers (" $\leq_{K}$ ") and its stronger variant-containing an isomorphic ideal ("⊑ "). In particular, we are interested in ideals for which $\leq_{K} \Rightarrow ⊑$ for every ideal . We find examples of ideals with this property and show how this property can be used to reformulate some problems known from the literature in terms of the Katětov order instead of the order "⊑ " (and vice versa).

Rings of continuous functions vanishing at infinity

Ali Rezaei Aliabad, F. Azarpanah, M. Namdari (2004)

Commentationes Mathematicae Universitatis Carolinae

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We prove that a Hausdorff space $X$ is locally compact if and only if its topology coincides with the weak topology induced by $C_{\infty} (X)$ . It is shown that for a Hausdorff space $X$ , there exists a locally compact Hausdorff space $Y$ such that $C_{\infty} (X) ≅ C_{\infty} (Y)$ . It is also shown that for locally compact spaces $X$ and $Y$ , $C_{\infty} (X) ≅ C_{\infty} (Y)$ if and only if $X ≅ Y$ . Prime ideals in $C_{\infty} (X)$ are uniquely represented by a class of prime ideals in $C^{*} (X)$ . $\infty$ -compact spaces are introduced and it turns out that a locally compact space $X$ is $\infty$ -compact if and only...

An upper bound for the distance to finitely generated ideals in Douglas algebras

Pamela Gorkin, Raymond Mortini, Daniel Suárez (2001)

Studia Mathematica

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Let f be a function in the Douglas algebra A and let I be a finitely generated ideal in A. We give an estimate for the distance from f to I that allows us to generalize a result obtained by Bourgain for $H^{\infty}$ to arbitrary Douglas algebras.

More on the strongly 1-absorbing primary ideals of commutative rings

Ali Yassine, Mohammad Javad Nikmehr, Reza Nikandish (2024)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with identity. We study the concept of strongly 1-absorbing primary ideals which is a generalization of $n$ -ideals and a subclass of $1$ -absorbing primary ideals. A proper ideal $I$ of $R$ is called strongly 1-absorbing primary if for all nonunit elements $a, b, c \in R$ such that $a b c \in I$ , it is either $a b \in I$ or $c \in \sqrt{0}$ . Some properties of strongly 1-absorbing primary ideals are studied. Finally, rings $R$ over which every semi-primary ideal is strongly 1-absorbing primary, and rings $R$ over which...

On norm closed ideals in $L (ℓ_{p}, ℓ_{q})$

B. Sari, Th. Schlumprecht, N. Tomczak-Jaegermann, V. G. Troitsky (2007)

Studia Mathematica

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It is well known that the only proper non-trivial norm closed ideal in the algebra L(X) for $X = ℓ_{p}$ (1 ≤ p < ∞) or X = c₀ is the ideal of compact operators. The next natural question is to describe all closed ideals of $L (ℓ_{p} \oplus ℓ_{q})$ for 1 ≤ p,q < ∞, p ≠ q, or equivalently, the closed ideals in $L (ℓ_{p}, ℓ_{q})$ for p < q. This paper shows that for 1 < p < 2 < q < ∞ there are at least four distinct proper closed ideals in $L (ℓ_{p}, ℓ_{q})$ , including one that has not been studied before. The proofs use various methods...

A primrose path from Krull to Zorn

Marcel Erné (1995)

Commentationes Mathematicae Universitatis Carolinae

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Given a set $X$ of “indeterminates” and a field $F$ , an ideal in the polynomial ring $R = F [X]$ is called conservative if it contains with any polynomial all of its monomials. The map $S \mapsto R S$ yields an isomorphism between the power set $P (X)$ and the complete lattice of all conservative prime ideals of $R$ . Moreover, the members of any system $S \subseteq P (X)$ of finite character are in one-to-one correspondence with the conservative prime ideals contained in $P_{S} = ⋃ {R S : S \in S}$ , and the maximal members of $S$ correspond to the maximal ideals contained...

Pretty cleanness and filter-regular sequences

Somayeh Bandari, Kamran Divaani-Aazar, Ali Soleyman Jahan (2014)

Czechoslovak Mathematical Journal

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Let $K$ be a field and $S = K [x_{1}, ..., x_{n}]$ . Let $I$ be a monomial ideal of $S$ and $u_{1}, ..., u_{r}$ be monomials in $S$ . We prove that if $u_{1}, ..., u_{r}$ form a filter-regular sequence on $S / I$ , then $S / I$ is pretty clean if and only if $S / (I, u_{1}, ..., u_{r})$ is pretty clean. Also, we show that if $u_{1}, ..., u_{r}$ form a filter-regular sequence on $S / I$ , then Stanley’s conjecture is true for $S / I$ if and only if it is true for $S / (I, u_{1}, ..., u_{r})$ . Finally, we prove that if $u_{1}, ..., u_{r}$ is a minimal set of generators for $I$ which form either a $d$ -sequence, proper sequence or strong $s$ -sequence (with respect to the reverse...