Krull–schmidt fails for artinian modules

Facchini Alberto; Herbera Dolors; Lawrence S. Levy; Peter Vámos

doi:10.1090/S0002-9939-1995-1277109-4

Krull–schmidt fails for artinian modules

Facchini Alberto, Herbera Dolors, Lawrence S. Levy, Peter Vámos

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

26 Citations (Scopus)

Abstract

We prove that the Krull-Schmidt theorem fails for artinian modules. This answers a question asked by Krull in 1932. In fact we show that if S is a module-finite algebra over a semilocal noetherian commutative ring, then every nonunique decomposition of every noetherian S-module leads to an analogous nonunique decomposition of an artinian module over a related non-noetherian ring. The key to this is that any such S is the endomorphism ring of some artinian module. © 1995 American Mathematical Society.

Original language	English
Pages (from-to)	3587-3592
Journal	Proceedings of the American Mathematical Society
Volume	123
Issue number	12
DOIs	https://doi.org/10.1090/S0002-9939-1995-1277109-4
Publication status	Published - 1 Jan 1995

Keywords

Capacities
Direct sum decomposition
Endomorphism ring
Krull-Schmidt

Access to Document

10.1090/S0002-9939-1995-1277109-4

Cite this

@article{8d3572cb54c34fadab5d05d6f2741e88,

title = "Krull–schmidt fails for artinian modules",

abstract = "We prove that the Krull-Schmidt theorem fails for artinian modules. This answers a question asked by Krull in 1932. In fact we show that if S is a module-finite algebra over a semilocal noetherian commutative ring, then every nonunique decomposition of every noetherian S-module leads to an analogous nonunique decomposition of an artinian module over a related non-noetherian ring. The key to this is that any such S is the endomorphism ring of some artinian module. {\textcopyright} 1995 American Mathematical Society.",

keywords = "Capacities, Direct sum decomposition, Endomorphism ring, Krull-Schmidt",

author = "Facchini Alberto and Herbera Dolors and Levy, {Lawrence S.} and Peter V{\'a}mos",

year = "1995",

month = jan,

day = "1",

doi = "10.1090/S0002-9939-1995-1277109-4",

language = "English",

volume = "123",

pages = "3587--3592",

journal = "Proceedings of the American Mathematical Society",

issn = "0002-9939",

publisher = "American Mathematical Society",

number = "12",

}

TY - JOUR

T1 - Krull–schmidt fails for artinian modules

AU - Alberto, Facchini

AU - Dolors, Herbera

AU - Levy, Lawrence S.

AU - Vámos, Peter

PY - 1995/1/1

Y1 - 1995/1/1

N2 - We prove that the Krull-Schmidt theorem fails for artinian modules. This answers a question asked by Krull in 1932. In fact we show that if S is a module-finite algebra over a semilocal noetherian commutative ring, then every nonunique decomposition of every noetherian S-module leads to an analogous nonunique decomposition of an artinian module over a related non-noetherian ring. The key to this is that any such S is the endomorphism ring of some artinian module. © 1995 American Mathematical Society.

AB - We prove that the Krull-Schmidt theorem fails for artinian modules. This answers a question asked by Krull in 1932. In fact we show that if S is a module-finite algebra over a semilocal noetherian commutative ring, then every nonunique decomposition of every noetherian S-module leads to an analogous nonunique decomposition of an artinian module over a related non-noetherian ring. The key to this is that any such S is the endomorphism ring of some artinian module. © 1995 American Mathematical Society.

KW - Capacities

KW - Direct sum decomposition

KW - Endomorphism ring

KW - Krull-Schmidt

U2 - 10.1090/S0002-9939-1995-1277109-4

DO - 10.1090/S0002-9939-1995-1277109-4

M3 - Article

SN - 0002-9939

VL - 123

SP - 3587

EP - 3592

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 12

ER -

Krull–schmidt fails for artinian modules

Abstract

Keywords

Access to Document

Fingerprint

Cite this