DECIDABILITY OF THE THEORY OF MODULES OVER PRÜFER DOMAINS WITH INFINITE RESIDUE FIELDS

AbstractWe provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Prüfer (in particular Bézout) domains with infinite residue fields in terms of a suitable generalization of the prime radical relation. For Bézout domains these conditions are also necessary.

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Bezout Domains and Rings with a Distributive Lattice of Right Ideals

Canadian Journal of Mathematics ◽

10.4153/cjm-1986-014-2 ◽

1986 ◽

Vol 38 (2) ◽

pp. 286-303 ◽

Cited By ~ 7

Author(s):

H. H. Brungs

Keyword(s):

Distributive Lattice ◽

Principal Ideal ◽

Commutative Rings ◽

Integral Domains ◽

Bezout Domain ◽

Prufer Domains ◽

Prüfer Domains ◽

Principal Ideal Domains ◽

Bezout Domains

It is the purpose of this paper to discuss a construction of right arithmetical (or right D-domains in [5]) domains, i.e., integral domains R for which the lattice of right ideals is distributive (see also [3]). Whereas the commutative rings in this class are precisely the Prüfer domains, not even right and left principal ideal domains are necessarily arithmetical. Among other things we show that a Bezout domain is right arithmetical if and only if all maximal right ideals are two-sided.Any right ideal of a right noetherian, right arithmetical domain is two-sided. This fact makes it possible to describe the semigroup of right ideals in such a ring in a satisfactory way; [3], [5].

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Decomposition and classification of length functions

Forum Mathematicum ◽

10.1515/forum-2018-0168 ◽

2020 ◽

Vol 32 (5) ◽

pp. 1109-1129

Author(s):

Dario Spirito

Keyword(s):

Integral Domain ◽

Integral Domains ◽

Bijective Correspondence ◽

Length Functions ◽

Standard Representation ◽

Prufer Domains ◽

Prüfer Domains

AbstractWe study decompositions of length functions on integral domains as sums of length functions constructed from overrings. We find a standard representation when the integral domain admits a Jaffard family, when it is Noetherian and when it is a Prüfer domains such that every ideal has only finitely many minimal primes. We also show that there is a natural bijective correspondence between singular length functions and localizing systems.

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Prüfer domains and pure submodules of direct sums of ideals

Mathematika ◽

10.1112/s0025579300007889 ◽

1999 ◽

Vol 46 (2) ◽

pp. 425-432 ◽

Cited By ~ 2

Author(s):

Bruce Olberding

Keyword(s):

Direct Sums ◽

Prufer Domains ◽

Pure Submodules ◽

Prüfer Domains

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On ideal theory in high prufer domains

manuscripta mathematica ◽

10.1007/bf01169264 ◽

1975 ◽

Vol 14 (4) ◽

pp. 303-336 ◽

Cited By ~ 2

Author(s):

Moshe Jarden

Keyword(s):

Ideal Theory ◽

Prufer Domains ◽

Prüfer Domains

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Arithmetic Rings and their generalizations

Bullettin of the Gioenia Academy of Natural Sciences of Catania ◽

10.35352/gioenia.v51i381.1 ◽

2018 ◽

Vol 51 (381) ◽

pp. FP1-FP6

Author(s):

R. Strano

Keyword(s):

Commutative Ring ◽

Natural Generalization ◽

Prüfer Domain ◽

Prufer Domains ◽

Prüfer Domains

Prüfer domains are characterized by various properties regarding ideals and operations between them. In this note we consider six of these properties. The natural generalization of the notion of Prüfer domain to the case of a commutative ring with unit, not necessarily a domain, is the notion of arithmetic ring. We ask if the previous properties characterize arithmetic ring in the case of a general commutative ring with unit. We prove that four of such properties characterize arithmetic rings while the remaining two are weaker and give rise to two different generalizations.

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