On generalized Dedekind prime rings

Akalan, Evrim

doi:10.1016/j.jalgebra.2008.07.001

Cited by 16 publications

(5 citation statements)

References 14 publications

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“…I personally do not think there is anything pseudo about the G-Dedekind domains. On the other hand some serious studies related to G-Dedekind prime rings, introduced by Evrim Akalan [9], are being carried out. This indicates that there is need for a name close to G-Dedekind domains.…”

Section: What's In a Name?mentioning

confidence: 99%

“…Theorem 3. 1 The following are equivalent for an integral domain R that is different from its quotient field K.…”

Section: Dually Compact Domainsmentioning

confidence: 99%

“…For the converse, note that (3) implies that R is at least a GCD domain and a GCD domain is Schreier. That is, (2) holds, and ( 2) is equivalent to (1). ∎…”

Section: Theorem 34 a Domain R Is A V-g-pid If And Only If R Is A DC ...mentioning

confidence: 99%

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Revisiting G-Dedekind domains

Zafrullah

2021

Can. Math. Bull.

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Let R be an integral domain with qf (R) = K and let F (R) be the set of nonzero fractional ideals of R. Call R a dually compact domain (DCD) if for each I 2 F (R) the ideal Iv = (I 1 ) 1 is a …nite intersection of principal fractional ideals. We characterize DCDs and show that the class of DCDs properly contains various classes of integral domains, such as Noetherian, Mori and Krull domains. In addition we show that a Schreier DCD is a GCD domain with the property that for each A 2 F (R) the ideal Av is principal. We show that a domain R is G-Dedekind (i.e. has the property that Av is invertible for each A 2 F (R)) if and only if R is a DCD satisfying the property : for all pairs of subsets fa 1 ; :::; amg; fb 1 ; :::; bng Knf0g;We discuss what the appropriate name for G-Dedekind domains and related notions should be. We also make some observations about how the DCDs behave under localizations and polynomial ring extensions.

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Section: What's In a Name?mentioning

confidence: 99%

“…Theorem 3. 1 The following are equivalent for an integral domain R that is different from its quotient field K.…”

Section: Dually Compact Domainsmentioning

confidence: 99%

See 1 more Smart Citation

Revisiting G-Dedekind domains

Zafrullah

2021

Can. Math. Bull.

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show abstract

“…I personally do not think there is anything pseudo about the G-Dedekind domains. On the other hand some serious studies related to G-Dedekind prime rings, introduced by Evrim Akalan [9], are being carried out. Hence the need for a name close to G-Dedekind domains.…”

Section: What's In a Name?mentioning

confidence: 99%

Revisiting G-Dedekind domains

Zafrullah¹

2021

Preprint

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Let R be an integral domain with qf (R) = K and let F (R) be the set of nonzero fractional ideals of R. Call R a dually compact domain (DCD) if for each I ∈ F (R) the ideal Iv = (I −1 ) −1 is a finite intersection of principal fractional ideals. We characterize DCDs and show that the class of DCDs properly contains various classes of integral domains, such as Noetherian, Mori and Krull domains. In addition we show that a Schreier DCD is a GCD domain with the property that for each A ∈ F (R) the ideal Av is principal. We show that a domain R is G-Dedekind domain (i.e. has the property that Av is invertible for each A ∈ F (R)) if and only if R is a DCD satisfying the property * : for all pairs of subsets {a 1 , ..., am}, {b 1 , ...bn} ⊆ K\{0},We discuss what the appropriate name for G-Dedekind domains and related notions should be. We also make some observations about how the DCDs behave under localizations and polynomial ring extensions.

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“…In [8], we defined a notion of a o-generalized Asano prime ring motivated by [1] and [2] Werefer readers to [9] or [10] for details of maximal orders and R-ideals. is a differential polynomial ring over / in an indeterminate x with multiplication xa = ax + 6(a), where is a derivation of R.…”

Section: Introductionmentioning

confidence: 99%