Let
A
be a (left and right) Noetherian ring that is semiperfect. Let
e
be an idempotent of
A
and consider the ring Γ:= (1 -
e
)
A
(1 -
e
) and the semi-simple right
A
-module
S
e
:=
e
A=
e
rad
A
. In ...this paper, we investigate the relationship between the global dimensions of
A
and Γ, by using the homological properties of
S
e
. More precisely, we consider the Yoneda ring
Y
(
e
):= Ext*
A
(
S
e
,
S
e
) of
e
. We prove that if
Y
(
e
) is Artinian of finite global dimension, then A has finite global dimension if and only if so does Γ. We also investigate the situation where both
A
and Γ have finite global dimension. When A is Koszul and finite dimensional, this implies that
Y
(
e
) has finite global dimension. We end the paper with a reduction technique to compute the Cartan determinant of Artin algebras. We prove that if
Y
(
e
) has finite global dimension, then the Cartan determinants of
A
and Γ coincide. This provides a new way to approach the long-standing Cartan determinant conjecture.
We prove a unique decomposition theorem for direct products of finitely generated modules over certain classes of rings, which is analogous to the classical Krull–Schmidt–Remak–Azumaya theorem for ...direct-sum decompositions of modules.
ON CLEAN LAURENT SERIES RINGS ZHOU, YIQIANG; ZIEMBOWSKI, MICHAŁ
Journal of the Australian Mathematical Society (2001),
12/2013, Volume:
95, Issue:
3
Journal Article
Peer reviewed
Open access
Here we prove that, for a $2$-primal ring $R$, the Laurent series ring $R((x))$ is a clean ring if and only if $R$ is a semiregular ring with $J(R)$ nil. This disproves the claim in K. I. Sonin ...‘Semiprime and semiperfect rings of Laurent series’, Math. Notes 60 (1996), 222–226 that the Laurent series ring over a clean ring is again clean. As an application of the result, it is shown that, for a $2$-primal ring $R$, $R((x))$ is semiperfect if and only if $R((x))$ is semiregular if and only if $R$ is semiperfect with $J(R)$ nil.
Classically, the Auslander-Bridger transpose finds its best applications in the well-known setting of finitely presented modules over a semiperfect ring. We introduce a class of modules over an ...arbitrary ring R, which we call Auslander-Bridger modules, with the property that the Auslander-Bridger transpose induces a well-behaved bijection between isomorphism classes of Auslander-Bridger right R-modules and isomorphism classes of Auslander-Bridger left R-modules. Thus we generalize what happens for finitely presented modules over a semiperfect ring. Auslander-Bridger modules are characterized by two invariants (epi-isomorphism class and lower-isomorphism class), which are interchanged by the transpose. Via a suitable duality, we find that kernels of morphisms between injective modules of finite Goldie dimension are also characterized by two invariants (mono-isomorphism class and upper-isomorphism class).
One of the main results of the article
2
says that, if a ring R is semiperfect and ϕ is an authomorphism of R, then the skew Laurent series ring R((x, ϕ)) is semiperfect. We will show that the ...above statement is not true. More precisely, we will show that, if the Laurent series ring R((x)) is semilocal, then R is semiperfect with nil Jacobson radical.
We initiate the study of 1-torsion of finite modules over two-sided noetherian semiperfect rings. In particular, we give a criterion for determining when the 1-torsion submodule contains minimal ...generators of the module. We also provide an explicit construction for a projective cover of the submodule generated by the torsion elements in the top of the module. Some of the obtained results hold without the noetherian assumption. We also give several applications to local algebra.
Let
be a ring and
a left
-module. An
-module
is called a
of
in case
and
is finitely generated. We say that
has
CE (resp.
CEE) if
has a supplement (resp. ample supplements) in every cofinite ...extension. In this study we give various properties of modules with these properties. We show that a module
has the property
CEE iff every submodule of
has the property
CE. A ring
is semiperfect iff every left
-module has the property
CE. We also study cofinitely injective modules, direct summands of every cofinite extension, as a generalization of injective modules.
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