## Almost slender rings and compact rings

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt

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**Almost slender rings and compact rings.** / Jensen, Chr Ulrik; Jøndrup, Søren; Thorup, Anders.

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt

#### Harvard

*Journal of Pure and Applied Algebra*, bind 223, nr. 5, s. 1869-1896. https://doi.org/10.1016/j.jpaa.2018.08.005

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*Journal of Pure and Applied Algebra*,

*223*(5), 1869-1896. https://doi.org/10.1016/j.jpaa.2018.08.005

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TY - JOUR

T1 - Almost slender rings and compact rings

AU - Jensen, Chr Ulrik

AU - Jøndrup, Søren

AU - Thorup, Anders

PY - 2019/5/1

Y1 - 2019/5/1

N2 - Classical results concerning slenderness for commutative integral domains are generalized to commutative rings with zero divisors. This is done by extending the methods from the domain case and bringing them in connection with results on the linear topologies associated to non-discrete Hausdorff filtrations. In many cases a weakened notion “almost slenderness” of slenderness is appropriate for rings with zero divisors. Special results for countable rings are extended to rings said to be of “bounded type” (including countable rings, ‘small’ rings, and, for instance, rings that are countably generated as algebras over an Artinian ring). More precisely, for a ring R of bounded type it is proved that R is slender if R is reduced and has no simple ideals, or if R is Noetherian and has no simple ideals; moreover, R is almost slender if R is not perfect (in the sense of H. Bass). We use our methods to study various special classes of rings, for instance von Neumann regular rings and valuation rings. Among other results we show that the following two rings are slender: the ring of Puiseux series over a field and the von Neumann regular ring kN/k(N) over a von Neumann regular ring k. For a Noetherian ring R we prove that R is a finite product of local complete rings iff R satisfies one of several (equivalent) conditions of algebraic compactness. A 1-dimensional Noetherian ring is outside this ‘compact’ class precisely when it is almost slender. For the rings of classical algebraic geometry we prove that a localization of an algebra finitely generated over a field is either Artinian or almost slender. Finally, we show that a Noetherian ring R is a finite product of local complete rings with finite residue fields exactly when there exists a map of R-algebras RN→R vanishing on R(N).

AB - Classical results concerning slenderness for commutative integral domains are generalized to commutative rings with zero divisors. This is done by extending the methods from the domain case and bringing them in connection with results on the linear topologies associated to non-discrete Hausdorff filtrations. In many cases a weakened notion “almost slenderness” of slenderness is appropriate for rings with zero divisors. Special results for countable rings are extended to rings said to be of “bounded type” (including countable rings, ‘small’ rings, and, for instance, rings that are countably generated as algebras over an Artinian ring). More precisely, for a ring R of bounded type it is proved that R is slender if R is reduced and has no simple ideals, or if R is Noetherian and has no simple ideals; moreover, R is almost slender if R is not perfect (in the sense of H. Bass). We use our methods to study various special classes of rings, for instance von Neumann regular rings and valuation rings. Among other results we show that the following two rings are slender: the ring of Puiseux series over a field and the von Neumann regular ring kN/k(N) over a von Neumann regular ring k. For a Noetherian ring R we prove that R is a finite product of local complete rings iff R satisfies one of several (equivalent) conditions of algebraic compactness. A 1-dimensional Noetherian ring is outside this ‘compact’ class precisely when it is almost slender. For the rings of classical algebraic geometry we prove that a localization of an algebra finitely generated over a field is either Artinian or almost slender. Finally, we show that a Noetherian ring R is a finite product of local complete rings with finite residue fields exactly when there exists a map of R-algebras RN→R vanishing on R(N).

UR - http://www.scopus.com/inward/record.url?scp=85052131481&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2018.08.005

DO - 10.1016/j.jpaa.2018.08.005

M3 - Journal article

AN - SCOPUS:85052131481

VL - 223

SP - 1869

EP - 1896

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 5

ER -

ID: 203664787