A Cartesian decomposition of a coherent configuration X is defined as a special set of its parabolics that form a Cartesian decomposition of the underlying set. It turns out that every tensor ...decomposition of X comes from a certain Cartesian decomposition. It is proved that if the coherent configuration X is thick, then there is a unique maximal Cartesian decomposition of X, i.e., there is exactly one internal tensor decomposition of X into indecomposable components. In particular, this implies an analog of the Krull-Schmidt theorem for the thick coherent configurations. A polynomial-time algorithm for finding the maximal Cartesian decomposition of a thick coherent configuration is constructed.
It is proved that two diagonal matrices
diag
(
a
1
,
…
,
a
n
)
and
diag
(
b
1
,
…
,
b
n
)
over a local ring
R are equivalent if and only if there are two permutations
σ
,
τ
of
{
1
,
2
,
…
,
n
}
such ...that
R
/
a
i
R
l
=
R
/
b
σ
(
i
)
R
l
and
R
/
a
i
R
e
=
R
/
b
τ
(
i
)
R
e
for every
i
=
1
,
2
,
…
,
n
. Here
R
/
a
R
e
denotes the epigeny class of
R
/
a
R
, and
R
/
a
R
l
denotes the lower part of
R
/
a
R
. In some particular cases, like for instance in the case of
R local commutative,
diag
(
a
1
,
…
,
a
n
)
is equivalent to
diag
(
b
1
,
…
,
b
n
)
if and only if there is a permutation
σ of
{
1
,
2
,
…
,
n
}
with
a
i
R
=
b
σ
(
i
)
R
for every
i
=
1
,
…
,
n
. These results are obtained studying the direct-sum decompositions of finite direct sums of cyclically presented modules over local rings. The theory of these decompositions turns out to be incredibly similar to the theory of direct-sum decompositions of finite direct sums of uniserial modules over arbitrary rings.
We say that an
R
-module
M
is
virtually semisimple
if each submodule of
M
is isomorphic to a direct summand of
M
. A nonzero indecomposable virtually semisimple module is then called a
virtually ...simple
module. We carry out a study of virtually semisimple modules and modules which are direct sums of virtually simple modules . Our study provides several natural generalizations of the Wedderburn-Artin Theorem and an analogous to the classical Krull-Schmidt Theorem. Some applications of these theorems are indicated. For instance, it is shown that the following statements are equivalent for a ring
R
: (i) Every finitely generated left (right)
R
-module is virtually semisimple; (ii) Every finitely generated left (right)
R
-module is a direct sum of virtually simple
R
-modules; (iii)
R
≅
∏
i
=
1
k
M
n
i
(
D
i
)
where
k
,
n
1
,…,
n
k
∈
ℕ
and each
D
i
is a principal ideal V-domain; and (iv) Every nonzero finitely generated left
R
-module can be written uniquely (up to isomorphism and order of the factors) in the form
R
m
1
⊕… ⊕
R
m
k
where each
R
m
i
is either a simple
R
-module or a virtually simple direct summand of
R
.
In this article we classify indecomposable objects of the derived categories of finitely-generated modules over certain infinite-dimensional algebras. The considered class of algebras (which we call ...nodal algebras) contains such well-known algebras as the complete ring of a double nodal point
k
x,y/(xy)
and the completed path algebra of the Gelfand quiver. As a corollary we obtain a description of the derived category of Harish-Chandra modules over
SL
2(
R)
. We also give an algorithm, which allows to construct projective resolutions of indecomposable complexes. In the appendix we prove the Krull–Schmidt theorem for homotopy categories.
We prove that the Krull-Schmidt Theorem holds for finite direct products of biuniform groups, that is, groups
G
whose lattice of normal subgroups
𝓝
(
G
)
has Goldie dimension and dual Goldie ...dimension 1. More generally, it holds for the class of completely indecomposable groups.
We study regularity in the infinite direct sum decomposition in cocomplete categories. In particular we investigate and characterize a weak form of the Krull-Schmidt-Azumaya Theorem,in which the ...uniqueness of the direct sum decomposition is granted up to not one but two bijections,providing an abstract setting for a behaviour observed in the category of serial modules and other categories.
The classes of uniserial modules, biuniform modules, cyclically presented modules over a local ring, more generally, couniformly presented modules, and kernels of morphisms between indecomposable ...injective modules, are some among the classes of modules which axe characterized by a pair of invariants. These invariants also completely describe when finite direct sums of such modules are isomorphic. In this paper, we are interested in modules characterized by finitely many invariants and in their finite direct sums. We give a general criterion to produce classes S of such modules, and we completely describe how modules satisfying said criterion can be grouped together to form isomorphic finite direct sums. The connection between the regularity of finite direct sums of modules in S and a certain associated hypergraph H(S) is also investigated.
We show that the indecomposable R-modules whose endomorphism ring has finitely many maximal right ideals, all of them two-sided, have a surprisingly simple behavior as far as direct sums are ...concerned. Our main result is that these modules are completely described up to isomorphism by an easy combinatorial structure, a simple hypergraph. If is any full subcategory of Mod-R containing all these modules as objects, the vertices of the hypergraph are suitable ideals of the category . Let SFT-R be the category of all finite direct sums of modules whose endomorphism ring has finitely many maximal right ideals. The objects of SFT-R are completely determined up to isomorphism by the dimensions of vector spaces indexed by suitable ideals of the category SFT-R. Several examples are given in the last section.
We show that every direct summand of a serial module
M of finite Goldie dimension is serial, give a description of the corresponding commutative monoid
V
(
M
)
, and generalize Facchini's weak ...Krull–Schmidt theorem to a larger class of modules.
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