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PROGRESS ON THE AUSLANDER–REITEN CONJECTURE
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 93 / Issue 3 / June 2016
- Published online by Cambridge University Press:
- 16 March 2016, pp. 433-440
- Print publication:
- June 2016
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- Article
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Let
$R$ be a commutative Gorenstein ring. A result of Araya reduces the Auslander–Reiten conjecture on the vanishing of self-extensions to the case where 
$R$ has Krull dimension at most one. In this paper we extend Araya’s result to certain 
$R$-algebras. As a consequence of our argument, we obtain examples of bound quiver algebras that satisfy the Auslander–Reiten conjecture.
On the Generalized Auslander–Reiten Conjecture under Certain Ring Extensions
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- Journal:
- Canadian Mathematical Bulletin / Volume 58 / Issue 1 / 01 March 2015
- Published online by Cambridge University Press:
- 20 November 2018, pp. 134-143
- Print publication:
- 01 March 2015
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We show that under some conditions a Gorenstein ring
$R$ satisfies the Generalized Auslander–Reiten conjecture if and only if 
$R\left[ x \right]$ does. When $R$ is a local ring we prove the same result for some localizations of $R\left[ x \right]$.
