# A family of noetherian rings with their finite length modules under control

Czechoslovak Mathematical Journal (2002)

- Volume: 52, Issue: 3, page 545-552
- ISSN: 0011-4642

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topSchmidmeier, Markus. "A family of noetherian rings with their finite length modules under control." Czechoslovak Mathematical Journal 52.3 (2002): 545-552. <http://eudml.org/doc/30723>.

@article{Schmidmeier2002,

abstract = {We investigate the category $\text\{mod\}\Lambda $ of finite length modules over the ring $\Lambda =A\otimes _k\Sigma $, where $\Sigma $ is a V-ring, i.e. a ring for which every simple module is injective, $k$ a subfield of its centre and $A$ an elementary $k$-algebra. Each simple module $E_j$ gives rise to a quasiprogenerator $P_j=A\otimes E_j$. By a result of K. Fuller, $P_j$ induces a category equivalence from which we deduce that $\text\{mod\}\Lambda \simeq \coprod _jbad hbox P_j$. As a consequence we can (1) construct for each elementary $k$-algebra $A$ over a finite field $k$ a nonartinian noetherian ring $\Lambda $ such that $\text\{mod\}A\simeq \text\{mod\}\Lambda $, (2) find twisted versions $\Lambda $ of algebras of wild representation type such that $\Lambda $ itself is of finite or tame representation type (in mod), (3) describe for certain rings $\Lambda $ the minimal almost split morphisms in $\text\{mod\} \Lambda $ and observe that almost all of these maps are not almost split in $\text\{Mod\}\Lambda $.},

author = {Schmidmeier, Markus},

journal = {Czechoslovak Mathematical Journal},

keywords = {V-ring; progenerator; almost split morphisms; V-rings; progenerators; almost split morphisms; categories of finite length modules; simple modules; category equivalences; Noetherian rings; wild representation type; finite representation type; tame representation type},

language = {eng},

number = {3},

pages = {545-552},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {A family of noetherian rings with their finite length modules under control},

url = {http://eudml.org/doc/30723},

volume = {52},

year = {2002},

}

TY - JOUR

AU - Schmidmeier, Markus

TI - A family of noetherian rings with their finite length modules under control

JO - Czechoslovak Mathematical Journal

PY - 2002

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 52

IS - 3

SP - 545

EP - 552

AB - We investigate the category $\text{mod}\Lambda $ of finite length modules over the ring $\Lambda =A\otimes _k\Sigma $, where $\Sigma $ is a V-ring, i.e. a ring for which every simple module is injective, $k$ a subfield of its centre and $A$ an elementary $k$-algebra. Each simple module $E_j$ gives rise to a quasiprogenerator $P_j=A\otimes E_j$. By a result of K. Fuller, $P_j$ induces a category equivalence from which we deduce that $\text{mod}\Lambda \simeq \coprod _jbad hbox P_j$. As a consequence we can (1) construct for each elementary $k$-algebra $A$ over a finite field $k$ a nonartinian noetherian ring $\Lambda $ such that $\text{mod}A\simeq \text{mod}\Lambda $, (2) find twisted versions $\Lambda $ of algebras of wild representation type such that $\Lambda $ itself is of finite or tame representation type (in mod), (3) describe for certain rings $\Lambda $ the minimal almost split morphisms in $\text{mod} \Lambda $ and observe that almost all of these maps are not almost split in $\text{Mod}\Lambda $.

LA - eng

KW - V-ring; progenerator; almost split morphisms; V-rings; progenerators; almost split morphisms; categories of finite length modules; simple modules; category equivalences; Noetherian rings; wild representation type; finite representation type; tame representation type

UR - http://eudml.org/doc/30723

ER -

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