2009/09/13 by Edgar E. Enochs, Enochs, Edgar E., Zhaoyong Huang +1
Mathematics · #16E10 #16E30 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #K-Theory and Homology (math.KT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.0909.2415
openalex publication_date 2009/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We characterize left Noetherian rings in terms of the duality property of injective preenvelopes and flat precovers. For a left and right Noetherian ring R, we prove that the flat dimension of the injective envelope of any (Gorenstein) flat left R-module is at most the flat dimension of the injective envelope of RR. Then we get that the injective envelope of RR is (Gorenstein) flat if and only if the injective envelope of every Gorenstein flat left R-module is (Gorenstein) flat, if and only if the injective envelope of every flat left R-module is (Gorenstein) flat, if and only if the (Gorenstein) flat cover of every injective left R-module is injective, and if and only if the opposite version of one of these conditions is satisfied.