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On Two Questions of M. Auslander

On Two Questions of M. Auslander SHEILA BRENNER For any ring R, let A (R) denote the ring of n x n matrices over R with zeros everywhere except on the principal diagonal and along the bottom row, where im (1) entries are chosen arbitrarily from R. Define A \R) by A (R) = A (R), n n a Auslander [1] has raised the questions (i) Is there an artin algebra R, with (rad R) = 0 and of finite representation type, but for which A (R) is not of finite representation type? (m) (ii) Is there an integer m such that, for every artin algebra R, A (R) is not of finite representation type? The answer to (i) is " yes " since, if A; is a field A (k) is of finite representation type if and only if n ^ 4 [2, 4] and A (A (A;)) has a quotient isomorphic to A (k). 2 4 5 The answer to (ii) is " yes, m = 4 " and we shall show that an even stronger result {m) holds for m = 5. First note that A (R) has a quotient isomorphic to A (R). 2 m+l If R is any commutative ring http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/4.3.301
Publisher site
See Article on Publisher Site

Abstract

SHEILA BRENNER For any ring R, let A (R) denote the ring of n x n matrices over R with zeros everywhere except on the principal diagonal and along the bottom row, where im (1) entries are chosen arbitrarily from R. Define A \R) by A (R) = A (R), n n a Auslander [1] has raised the questions (i) Is there an artin algebra R, with (rad R) = 0 and of finite representation type, but for which A (R) is not of finite representation type? (m) (ii) Is there an integer m such that, for every artin algebra R, A (R) is not of finite representation type? The answer to (i) is " yes " since, if A; is a field A (k) is of finite representation type if and only if n ^ 4 [2, 4] and A (A (A;)) has a quotient isomorphic to A (k). 2 4 5 The answer to (ii) is " yes, m = 4 " and we shall show that an even stronger result {m) holds for m = 5. First note that A (R) has a quotient isomorphic to A (R). 2 m+l If R is any commutative ring

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Nov 1, 1972

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