In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, a uniserial module ''M'' is a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
over a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
''R'', whose
submodule
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...
s are
totally ordered
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive ...
by
inclusion
Inclusion or Include may refer to:
Sociology
* Social inclusion, aims to create an environment that supports equal opportunity for individuals and groups that form a society.
** Inclusion (disability rights), promotion of people with disabiliti ...
. This means simply that for any two submodules ''N''
1 and ''N''
2 of ''M'', either
or
. A module is called a serial module if it is a
direct sum of uniserial modules. A ring ''R'' is called a right uniserial ring if it is uniserial as a right module over itself, and likewise called a right serial ring if it is a right serial module over itself. Left uniserial and left serial rings are defined in an analogous way, and are in general distinct from their right counterparts.
An easy motivating example is the
quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
for any
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
. This ring is always serial, and is uniserial when ''n'' is a
prime power
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number.
For example: , and are prime powers, while
, and are not.
The sequence of prime powers begins:
2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17 ...
.
The term ''uniserial'' has been used differently from the above definition: for clarification
see below.
A partial alphabetical list of important contributors to the theory of serial rings includes the mathematicians Keizo Asano, I. S. Cohen,
P.M. Cohn, Yu. Drozd,
D. Eisenbud, A. Facchini,
A.W. Goldie, Phillip Griffith,
I. Kaplansky, V.V Kirichenko,
G. Köthe, H. Kuppisch, I. Murase,
T. Nakayama, P. Příhoda, G. Puninski, and R. Warfield. References for each author can be found in and .
Following the common ring theoretic convention, if a left/right dependent condition is given without mention of a side (for example, uniserial, serial,
Artinian,
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
) then it is assumed the condition holds on both the left and right. Unless otherwise specified, each ring in this article is a
ring with unity, and each module is
unital.
Properties of uniserial and serial rings and modules
It is immediate that in a uniserial ''R''-module ''M'', all submodules except ''M'' and 0 are simultaneously
essential and
superfluous. If ''M'' has a
maximal submodule
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
, then ''M'' is a
local module
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
* Local government, a form of public administration, usually the lowest tier of administrat ...
. ''M'' is also clearly a
uniform module In abstract algebra, a module is called a uniform module if the intersection of any two nonzero submodules is nonzero. This is equivalent to saying that every nonzero submodule of ''M'' is an essential submodule. A ring may be called a right (lef ...
and thus is directly indecomposable. It is also easy to see that every finitely generated submodule of ''M'' can be generated by a single element, and so ''M'' is a Bézout module.
It is known that the
endomorphism ring
In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a ...
End
R(''M'') is a
semilocal ring
In mathematics, a semi-local ring is a ring for which ''R''/J(''R'') is a semisimple ring, where J(''R'') is the Jacobson radical of ''R''.
The above definition is satisfied if ''R'' has a finite number of maximal right ideals (and finite numb ...
which is very close to a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
in the sense that End
R(''M'') has at most two
maximal right ideals. If ''M'' is assumed to be Artinian or Noetherian, then End
R(''M'') is a local ring.
Since rings with unity always have a maximal right ideal, a right uniserial ring is necessarily local. As noted before, a finitely generated right ideal can be generated by a single element, and so right uniserial rings are right Bézout rings. A right serial ring ''R'' necessarily factors in the form
where each ''e''
i is an
idempotent element
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
and ''e''
i''R'' is a local, uniserial module. This indicates that ''R'' is also a
semiperfect ring, which is a stronger condition than being a semilocal ring.
Köthe showed that the modules of Artinian
principal ideal ring In mathematics, a principal right (left) ideal ring is a ring ''R'' in which every right (left) ideal is of the form ''xR'' (''Rx'') for some element ''x'' of ''R''. (The right and left ideals of this form, generated by one element, are called prin ...
s (which are a special case of serial rings) are direct sums of
cyclic submodules. Later, Cohen and Kaplansky determined that a
commutative ring ''R'' has this property for its modules if and only if ''R'' is an Artinian principal ideal ring. Nakayama showed that Artinian serial rings have this property on their modules, and that the converse is not true
The most general result, perhaps, on the modules of a serial ring is attributed to Drozd and Warfield: it states that every
finitely presented module over a serial ring is a direct sum of cyclic uniserial submodules (and hence is serial). If additionally the ring is assumed to be Noetherian, the finitely presented and finitely generated modules coincide, and so all finitely generated modules are serial.
Being right serial is preserved under direct products of rings and modules, and preserved under
quotients of rings. Being uniserial is preserved for quotients of rings and modules, but never for products. A direct summand of a serial module is not necessarily serial, as was proved by Puninski, but direct summands of ''finite'' direct sums of uniserial modules are serial modules .
It has been verified that
Jacobson's conjecture holds in Noetherian serial rings.
Examples
Any
simple module In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every cy ...
is trivially uniserial, and likewise
semisimple module
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itsel ...
s are serial modules.
Many examples of serial rings can be gleaned from the structure sections above. Every
valuation ring In abstract algebra, a valuation ring is an integral domain ''D'' such that for every element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x''−1 belongs to ''D''.
Given a field ''F'', if ''D'' is a subring of ''F'' suc ...
is a uniserial ring, and all Artinian principal ideal rings are serial rings, as is illustrated by
semisimple ring
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itsel ...
s.
More exotic examples include the
upper triangular matrices
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are ...
over a
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
T
''n''(''D''), and the
group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the giv ...