2 results
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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- Article
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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 $R\left[ x \right]$.
Existentially closed models of the theory of artinian local rings
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- Journal:
- The Journal of Symbolic Logic / Volume 64 / Issue 2 / June 1999
- Published online by Cambridge University Press:
- 12 March 2014, pp. 825-845
- Print publication:
- June 1999
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The class of all Artinian local rings of length at most l is ∀2-elementary, axiomatised by a finite set of axioms
τtl. We show that its existentially closed models are Gorenstein. of length exactly l and their residue fields are algebraically closed, and, conversely, every existentially closed model is of this form. The theory 
oτl of all Artinian local Gorenstein rings of length l with algebraically closed residue field is model complete and the theory τtl is companionable, with model-companion oτl.
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