Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Injective Envelopes and (Gorenstein) Flat Covers   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.

3
The categories of this paper are: physics
23
The categories of this paper are: solv-int
31
The categories of this paper are: ao-sci
29
The categories of this paper are: funct-an
24
The categories of this paper are: dg-ga
22
The categories of this paper are: chao-dyn
20
The categories of this paper are: acc-phys
0
The categories of this paper are: hep-ph
33
The categories of this paper are: plasm-ph
35
The categories of this paper are: mtrl-th
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
19
The categories of this paper are: eess
13
The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex