Integral domains in which any two v-coprime elements are comaximal

Houston, Evan; Zafrullah, Muhammad

doi:10.1016/j.jalgebra.2014.10.006

Cited by 7 publications

(3 citation statements)

References 27 publications

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“…Example 2.4. Let R be an integral domain which is not DW (for a concrete example, see [10,Proposition 5.2]). Take a proper GV-ideal J of R. Since R is a w-ideal, it follows from [26,Theorem 6.1.14]…”

Section: Resultsmentioning

confidence: 99%

Baer submodules of modules over commutative rings

Anebri

Kim

Mahdou

2023

International Electronic Journal of Algebra

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Let $R$ be a commutative ring and $M$ be an $R$-module. A submodule $N$ of $M$ is called a d-submodule $($resp., an fd-submodule$)$ if $\ann_R(m)\subseteq \ann_R(m')$ $($resp., $\ann_R(F)\subseteq \ann_R(m'))$ for some $m\in N$ $($resp., finite subset $F\subseteq N)$ and $m'\in M$ implies that $m'\in N.$ Many examples, characterizations, and properties of these submodules are given. Moreover, we use them to characterize modules satisfying Property T, reduced modules, and von Neumann regular modules.

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Section: Resultsmentioning

confidence: 99%

Baer submodules of modules over commutative rings

Anebri

Kim

Mahdou

2023

International Electronic Journal of Algebra

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“…We close this section with the next result, which features necessary conditions for such a coincidence. For this purpose, recall that a domain where the trivial and w-operations are the same is said to be a DW-domain [16,25,31]. (2) Assume that R is integrally closed and let I be a finitely generated ideal of R and J a t-reduction of I.…”

Section: Proposition 24 Let R Be a Pseudo-valuation Domain And Let J ...mentioning

confidence: 99%

On t-reductions of ideals in pullbacks

Kabbaj,

Kadri,

Mimouni

2016

Preprint

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Let R be an integral domain and I a nonzero ideal of R. An ideal J ⊆ I is a t-reduction of I if (JI n ) t = (I n+1 ) t for some positive integer n; and I is t-basic if it has no t-reduction other than the trivial ones. This paper investigates t-reductions of ideals in pullback constructions of type . Section 2 examines the correlation between the notions of reduction and t-reduction in pseudo-valuation domains. Section 3 solves an open problem on whether the finite t-basic and v-basic ideal properties are distinct. We prove that these two notions coincide in any arbitrary domain. Section 4 features the main result, which establishes the transfer of the finite t-basic ideal property to pullbacks in line with Fontana-Gabelli's result on PvMDs [9, Theorem 4.1] and Gabelli-Houston's result on v-domains [14, Theorem 4.15]. This allows us to enrich the literature with new families of examples, which put the class of domains subject to the finite t-basic ideal property strictly between the two classes of v-domains and integrally closed domains.

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“…For instance, a domain is Pr üfer (resp., a PvMD) if every nonzero finitely generated ideal is invertible (resp., tinvertible). For some relevant works on v-and t-operations, we refer the reader to [10,16,19,20,21,24,25,26,27].…”

Section: Introductionmentioning

confidence: 99%

$t$-reductions and $t$-integral closure of ideals

Kabbaj¹,

Kadri²

2017

Rocky Mountain J. Math.

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Let R be an integral domain and I a nonzero ideal of R. An ideal J ⊆ I is a t-reduction of I if (JI n ) t = (I n+1 ) t for some integer n ≥ 0. An element x ∈ R is t-integral over I if there is an equation x n + a 1 x n−1 + ... + a n−1 x + a n = 0 with a i ∈ (I i ) t for i = 1, ..., n. The set of all elements that are t-integral over I is called the tintegral closure of I. This paper investigates the t-reductions and t-integral closure of ideals. Our objective is to establish satisfactory t-analogues of well-known results, in the literature, on the integral closure of ideals and its correlation with reductions. Namely, Section 2 identifies basic properties of t-reductions of ideals and features explicit examples discriminating between the notions of reduction and t-reduction. Section 3 investigates the concept of t-integral closure of ideals, including its correlation with t-reductions. Section 4 studies the persistence and contraction of t-integral closure of ideals under ring homomorphisms. All along the paper, the main results are illustrated with original examples. 1 t-Reductions of idealsThis section identifies basic ideal-theoretic properties of the notion of t-reduction including its behavior under localizations. As a prelude to this, we provide explicit examples discriminating between the notions of reduction and t-reduction.Recall that, in a ring R, a subideal J of an ideal I is called a reduction of I if JI n = I n+1 for some positive integer n [23]. An ideal which has no reduction other than itself is called a basic ideal [12,13].Definition 2.1 (cf. [15, Definition 1.1]). Let R be a domain and I a nonzero ideal of R. An ideal J ⊆ I is a t-reduction of I if (JI n ) t = (I n+1 ) t for some integer n ≥ 0 (and, a fortiori, the relation holds for n ≫ 0). The ideal J is a trivial t-reduction of I if J t = I t . The ideal I is t-basic if it has no t-reduction other than the trivial t-reductions.At this point, recall a basic property of the t-operation (which, in fact, holds for any star operation) that will be used throughout the paper. For any two nonzero ideals I and J of a domain, we have (IJ) t = (I t J) t = (IJ t ) t = (I t J t ) t . So, obviously, for nonzero ideals J ⊆ I, we always have:Notice also that a reduction is necessarily a t-reduction; and the converse is not true, in general, as shown by the next example which exhibits a domain R with two t-ideals J I such that J is a t-reduction but not a reduction of I. Example 2.2. We use a construction from [18]. Let x be an indeterminate over Z and let R := Z[3x, x 2 , x 3 ], I := (3x, x 2 , x 3 ), and J := (3x, 3x 2 , x 3 , x 4 ). Then J I are two finitely generated t-ideals of R such that: JI n I n+1 ∀ n ∈ N and (JI) t = (I 2 ) t .

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Integral domains in which any two v-coprime elements are comaximal

Cited by 7 publications

References 27 publications

Baer submodules of modules over commutative rings

Baer submodules of modules over commutative rings

On t-reductions of ideals in pullbacks

$t$-reductions and $t$-integral closure of ideals

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