Products of elementary and idempotent matrices over integral domains

Salce, Luigi; Zanardo, Paolo

doi:10.1016/j.laa.2014.03.042

Cited by 14 publications

(45 citation statements)

References 17 publications

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“…The above results motivated Salce and Zanardo [22] (2014) to conjecture that an integral domain satisfying ID 2 must be a Bézout domain. The classes of unique factorization domains, projective-free domains, and PRINC domains (defined in [22], and later studied in [7,20]) verify the conjecture (cf. [4,22]).…”

Section: Introductionmentioning

confidence: 67%

“…An easy computation shows that a singular nonzero matrix a b c d ∈ M 2 (R) is idempotent if and only if d = 1 − a (cf. [22]). A pair of elements a, b ∈ R is said to be an idempotent pair if (a b) is the first row of an idempotent matrix.…”

Section: Notation and Preliminary Resultsmentioning

confidence: 99%

“…We will mainly consider matrices of the form x y 0 0 , x, y ∈ R. Hence, to simplify the notation, we will denote by [x y] the singular matrix x y 0 0 . Following the notation in [22], we say that two nonzero elements a, b of an integral domain R admit a weak (Euclidean) algorithm if there exists a sequence of divisions r i = q i+1 r i+1 + r i+2 , with r i , q i ∈ R, −1 ≤ i ≤ n − 2, such that a = r −1 , b = r 0 and r n = 0.…”

Section: Notation and Preliminary Resultsmentioning

confidence: 99%

“…An easy computation shows that if M is a singular matrix in M 2 (R) and the ideal generated by the elements of its first row is principal, then M is a column-row matrix [22,Prop. 2.2].…”

Section: Column-row Matrices and Examplesmentioning

confidence: 99%

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Idempotent factorizations of singular 2 × 2 matrices over quadratic integer rings

Cossu

Zanardo

2020

Linear and Multilinear Algebra

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Let D be the ring of integers of a quadratic number fieldWe study the factorizations of 2 × 2 matrices over D into idempotent factors. When d < 0 there exist singular matrices that do not admit idempotent factorizations, due to results by Cohn (1965) and by the authors (2019). We mainly investigate the case d > 0. We employ Vaseršteȋn's result (1972) that SL2(D) is generated by elementary matrices, to prove that any 2 × 2 matrix with either a null row or a null column is a product of idempotents. As a consequence, every columnrow matrix admits idempotent factorizations.

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Section: Introductionmentioning

confidence: 67%

Section: Notation and Preliminary Resultsmentioning

confidence: 99%

Section: Notation and Preliminary Resultsmentioning

confidence: 99%

“…An easy computation shows that if M is a singular matrix in M 2 (R) and the ideal generated by the elements of its first row is principal, then M is a column-row matrix [22,Prop. 2.2].…”

Section: Column-row Matrices and Examplesmentioning

confidence: 99%

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Idempotent factorizations of singular 2 × 2 matrices over quadratic integer rings

Cossu

Zanardo

2020

Linear and Multilinear Algebra

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“…Various aspects of factorization theory could not be covered in this survey. These include factorizations in non-commutative rings and semigroups ( [86]), factorizations in commutative rings with zero-divisors ( [5]), arithmetic of non-atomic, non-BF, and non-Mori domains ( [8,23,24]), and factorizations into distinguished elements that are not irreducible (e.g., factorizations into radical ideals and others [32,81,75,76]).…”

Section: Introductionmentioning

confidence: 99%

Factorization theory in commutative monoids

Geroldinger

Zhong

2020

Semigroup Forum

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This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and their catenary degrees. We also provide examples where these finiteness properties do not hold.

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Idempotent Pairs and PRINC Domains

Peruginelli

Salce

Zanardo

2016

Springer Proceedings in Mathematics &Amp; Statistics

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A pair of elements a, b in an integral domain R is an idempotent pair if either a(1 − a) ∈ bR, or b(1 − b) ∈ aR. R is said to be a PRINC domain if all the ideals generated by an idempotent pair are principal. We show that in an order R of a Dedekind domain every regular prime ideal can be generated by an idempotent pair; moreover, if R is PRINC, then its integral closure, which is a Dedekind domain, is PRINC, too. Hence, a Dedekind domain is PRINC if and only if it is a PID. Furthermore, we show that the only imaginary quadratic orders Z[ √ −d], d > 0 square-free, that are PRINC and not integrally closed, are for d = 3, 7.

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Products of elementary and idempotent matrices over integral domains

Cited by 14 publications

References 17 publications

Idempotent factorizations of singular 2 × 2 matrices over quadratic integer rings

Idempotent factorizations of singular 2 × 2 matrices over quadratic integer rings

Factorization theory in commutative monoids

Idempotent Pairs and PRINC Domains

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