Analogs of Jacobian conditions for subrings

Jędrzejewicz, Piotr; Zieliński, J.

doi:10.48550/arxiv.1601.01508

Cited by 3 publications

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“…The last section contains various properties in a factorial form. In the first proposition we express a condition from [12], Theorem 3.4 in terms of irreducible and square-free factorizations. Proposition 4.1.…”

Section: Some Factorial Conditions For Subringsmentioning

confidence: 99%

“…Theorem ( [12], 3.4) Let A be a unique factorization domain. Let R be a subring of A such that R * = A * and R 0 ∩ A = R. The following conditions are equivalent:…”

Section: Introductionmentioning

confidence: 99%

“…A subring R satisfying the above condition (ii) is called square-factorially closed in A ( [12], Definition 3.5). Under the assumptions of Theorem 3.4 square-factorially closed subrings are root closed ( [12], Theorem 3.6), see [1] and [5] for information on root closed subrings.…”

Section: Introductionmentioning

confidence: 99%

“…A general question stated in [12], when do conditions (ii) or (iii) of Theorem 2.4 imply algebraic closedness of R in A, inspired us to study their relations with other conditions of this type (see Proposition 3.3 below). The above Theorem 3.4 motivates us to investigate various properties having a form of factoriality, in particular similar to (ii).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On some factorial properties of subrings

Jędrzejewicz¹,

Matysiak²,

Zieliński³

2017

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Section: Some Factorial Conditions For Subringsmentioning

confidence: 99%

“…Theorem ( [12], 3.4) Let A be a unique factorization domain. Let R be a subring of A such that R * = A * and R 0 ∩ A = R. The following conditions are equivalent:…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On some factorial properties of subrings

Jędrzejewicz¹,

Matysiak²,

Zieliński³

2017

Self Cite

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“…A subring R of a UFD A is called square-factorially closed in A, if for every a ∈ A and square-free b ∈ A (see [21, page 5]), a 2 b ∈ R − 0 implies that a, b ∈ R − 0; see [21,Definition 3.5].…”

Section: Preliminariesmentioning

confidence: 99%

Ideas about the Jacobian Conjecture

Moskowicz¹

2016

Preprint

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Let F : C[x 1 , . . . , xn] → C[x 1 , . . . , xn] be a C-algebra endomorphism that has an invertible Jacobian.We bring two ideas concerning the Jacobian Conjecture: First, we conjecture that for all n, the degree of the field extension C(F (x 1 ), . . . , F (xn)) ⊆ C(x 1 , . . . , xn) is less than or equal to d n−1 , where d is the minimum of the degrees of the F (x i )'s. If this conjecture is true, then the generalized Jacobian Conjecture is true.Second, we suggest to replace in some known theorems the assumption on the degrees of the F (x i )'s by a similar assumption on the degrees of the minimal polynomials of the x i 's over C(F (x 1 ), . . . , F (xn)); this way we obtain some analogous results to the known ones.2010 Mathematics Subject Classification. Primary 14R15.[42] Y. Zhang, The Jacobian Conjecture and the degree of field extension, PhD thesis, Purdue University, 1991.

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On properties of square-free elements in commutative cancellative monoids

et al. 2019

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We discuss various square-free factorizations in monoids in the context of: atomicity, ascending chain condition for principal ideals, decomposition, and a greatest common divisor property. Moreover, we obtain a full characterization of submonoids of factorial monoids in which all square-free elements of a submonoid are square-free in a monoid. We also present factorial properties implying that all atoms of a submonoid are square-free in a monoid.(1.1) A(M) ⊂ S(H).

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Analogs of Jacobian conditions for subrings

Cited by 3 publications

References 0 publications

On some factorial properties of subrings

On some factorial properties of subrings

Ideas about the Jacobian Conjecture

On properties of square-free elements in commutative cancellative monoids

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