ω 1 -generated uniserial modules over chain rings

Jan Žemlička

Commentationes Mathematicae Universitatis Carolinae (2004)

  • Volume: 45, Issue: 3, page 403-415
  • ISSN: 0010-2628

Abstract

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The purpose of this paper is to provide a criterion of an occurrence of uncountably generated uniserial modules over chain rings. As we show it suffices to investigate two extreme cases, nearly simple chain rings, i.e. chain rings containing only three two-sided ideals, and chain rings with “many” two-sided ideals. We prove that there exists an ω 1 -generated uniserial module over every non-artinian nearly simple chain ring and over chain rings containing an uncountable strictly increasing (resp. decreasing) chain of right (resp. two-sided) ideals. As a consequence we describe right steady serial rings.

How to cite

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Žemlička, Jan. "$\omega _1$-generated uniserial modules over chain rings." Commentationes Mathematicae Universitatis Carolinae 45.3 (2004): 403-415. <http://eudml.org/doc/249372>.

@article{Žemlička2004,
abstract = {The purpose of this paper is to provide a criterion of an occurrence of uncountably generated uniserial modules over chain rings. As we show it suffices to investigate two extreme cases, nearly simple chain rings, i.e. chain rings containing only three two-sided ideals, and chain rings with “many” two-sided ideals. We prove that there exists an $\omega _\{1\}$-generated uniserial module over every non-artinian nearly simple chain ring and over chain rings containing an uncountable strictly increasing (resp. decreasing) chain of right (resp. two-sided) ideals. As a consequence we describe right steady serial rings.},
author = {Žemlička, Jan},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {chain rings; serial rings; uniserial modules; serial rings; uncountably generated uniserial modules; simple chain rings; lattices of ideals; right steady rings; dually slender modules},
language = {eng},
number = {3},
pages = {403-415},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {$\omega _1$-generated uniserial modules over chain rings},
url = {http://eudml.org/doc/249372},
volume = {45},
year = {2004},
}

TY - JOUR
AU - Žemlička, Jan
TI - $\omega _1$-generated uniserial modules over chain rings
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2004
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 45
IS - 3
SP - 403
EP - 415
AB - The purpose of this paper is to provide a criterion of an occurrence of uncountably generated uniserial modules over chain rings. As we show it suffices to investigate two extreme cases, nearly simple chain rings, i.e. chain rings containing only three two-sided ideals, and chain rings with “many” two-sided ideals. We prove that there exists an $\omega _{1}$-generated uniserial module over every non-artinian nearly simple chain ring and over chain rings containing an uncountable strictly increasing (resp. decreasing) chain of right (resp. two-sided) ideals. As a consequence we describe right steady serial rings.
LA - eng
KW - chain rings; serial rings; uniserial modules; serial rings; uncountably generated uniserial modules; simple chain rings; lattices of ideals; right steady rings; dually slender modules
UR - http://eudml.org/doc/249372
ER -

References

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  10. Puninski G.E., Tuganbaev A.A., Rings and Modules (Kol'ca i moduli), Moskva, 1998 (in Russian). MR1641739
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  12. Trlifaj J., Almost ☆ -modules need not be finitely generated, Comm. Algebra (1993), 21 2453-2462. (1993) MR1218507
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