Affiliation:
1. Department of Mathematics , Nanjing University , Nanjing 210093, Jiangsu Province , P. R. China
Abstract
Abstract
Let R and S be arbitrary rings and let
C
S
R
{{}_{R}C_{S}}
be a semidualizing bimodule, and let
𝒜
C
(
R
op
)
{\mathcal{A}_{C}(R^{\mathrm{op}})}
and
ℬ
C
(
R
)
{\mathcal{B}_{C}(R)}
be the Auslander and Bass classes, respectively. Then both pairs
(
𝒜
C
(
R
op
)
,
ℬ
C
(
R
)
)
and
(
ℬ
C
(
R
)
,
𝒜
C
(
R
op
)
)
(\mathcal{A}_{C}(R^{\mathrm{op}}),\mathcal{B}_{C}(R))\quad\text{and}\quad(%
\mathcal{B}_{C}(R),\mathcal{A}_{C}(R^{\mathrm{op}}))
are coproduct-closed and product-closed duality pairs and both
𝒜
C
(
R
op
)
{\mathcal{A}_{C}(R^{\mathrm{op}})}
and
ℬ
C
(
R
)
{\mathcal{B}_{C}(R)}
are covering and preenveloping;
in particular, the former duality pair is perfect. Moreover,
if
ℬ
C
(
R
)
{\mathcal{B}_{C}(R)}
is enveloping in
Mod
R
{\operatorname{Mod}R}
, then
𝒜
C
(
S
)
{\mathcal{A}_{C}(S)}
is enveloping in
Mod
S
{\operatorname{Mod}S}
.
Also, some applications to the Auslander projective dimension of modules are given.
Cited by
2 articles.
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