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| We prove that for a left and right Noetherian ring $R$, $_RR$satisfies the Auslander condition if and only if so does every flatleft $R$-module, if and only if the injective dimension of the $i$thterm in a minimal flat resolution of any injective left $R$-moduleis at most $i-1$ for any $i geq 1$, if and only if the flat (resp.injective) dimension of the $i$th term in a minimal injective (resp.flat) resolution of any left $R$-module $M$ is at most the flat(resp. injective) dimension of $M$ plus $i-1$ for any $i geq 1$, ifand only if the flat (resp. injective) dimension of the injectiveenvelope (resp. flat cover) of any left $R$-module $M$ is at mostthe flat (resp. injective) dimension of $M$, and if and only if anyof the opposite versions of the above conditions hold true.Furthermore, we prove that for an Artinian algebra $R$ satisfyingthe Auslander condition, $R$ is Gorenstein if and only if thesubcategory consisting of finitely generated modules satisfying theAuslander condition is contravariantly finite.As applications, we get some equivalentcharacterizations of Auslander-Gorenstein rings andAuslander-regular rings. |
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| Keywords:Pure Mathematics, Auslander-type conditions, Flat dimension, Injective dimension, Minimal flatresolutions, Minimal injective coresolutions, Gorenstein algebras,Contravariantly finite subcategories. |
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