Azumaya's Theorem
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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 decomposition of a module is a way to write 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 ...
as a
direct sum of modules In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, ma ...
. A type of a decomposition is often used to define or characterize modules: for example, a
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 ...
is a module that has a decomposition into
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 ...
s. Given 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 ...
, the types of decomposition of modules over the ring can also be used to define or characterize the ring: a ring is semisimple
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...
every module over it is a semisimple module. An
indecomposable module In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules. Jacobson (2009), p. 111. Indecomposable is a weaker notion than simple module (which is also sometimes called irr ...
is a module that is not a direct sum of two nonzero
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. Azumaya's theorem states that if a module has an decomposition into modules with
local 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 ...
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 ...
s, then all decompositions into indecomposable modules are equivalent to each other; a special case of this, especially in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, is known as the
Krull–Schmidt theorem In mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups. Definitions We say that a group ''G ...
. A special case of a decomposition of a module is a decomposition of a ring: for example, a ring is semisimple if and only if it is a direct sum (in fact a
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
) of matrix rings over
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 ...
s (this observation is known as the Artin–Wedderburn theorem).


Idempotents and decompositions

To give a direct sum decomposition of a module into submodules is the same as to give orthogonal idempotents in 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 ...
of the module that sum up to the
identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
. Indeed, if M = \bigoplus_ M_i, then, for each i \in I, the
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...
e_i : M \to M_i \hookrightarrow M given by the natural projection followed by the natural inclusion is an
idempotent 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 ...
. They are clearly orthogonal to each other (e_i e_j = 0 for i \ne j) and they sum up to the identity map: :1_ = \sum_ e_i as endomorphisms (here the summation is well-defined since it is a finite sum at each element of the module). Conversely, each set of orthogonal idempotents \_ such that only finitely many e_i(x) are nonzero for each x \in M and images An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of e_i. This fact already puts some constraints on a possible decomposition of a ring: given a ring R, suppose there is a decomposition :_R R = \bigoplus_ I_a of R as a left module over itself, where I_a are left submodules; i.e., left ideals. Each endomorphism _R R \to _R R can be identified with a right multiplication by an element of ''R''; thus, I_a = R e_a where e_a are idempotents of \operatorname(_R R) \simeq R. The summation of idempotent endomorphisms corresponds to the decomposition of the unity of ''R'': 1_R = \sum_ e_a \in \bigoplus_ I_a, which is necessarily a finite sum; in particular, A must be a finite set. For example, take R = \operatorname_n(D), the ring of ''n''-by-''n''
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
over a division ring ''D''. Then _R R is the direct sum of ''n'' copies of D^n, the columns; each column is a simple left ''R''-submodule or, in other words, a minimal left ideal. Let ''R'' be a ring. Suppose there is a (necessarily finite) decomposition of it as a left module over itself :_R R = R_1 \oplus \cdots \oplus R_n into ''two-sided ideals'' R_i of ''R''. As above, R_i = R e_i for some orthogonal idempotents e_i such that \textstyle. Since R_i is an ideal, e_i R \subset R_i and so e_i R e_j \subset R_i \cap R_j = 0 for i \ne j. Then, for each ''i'', :e_i r = \sum_j e_j r e_i = \sum_j e_i r e_j = r e_i. That is, the e_i are in the
center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...
; i.e., they are
central idempotent In ring theory, a branch of abstract algebra, an idempotent element or simply idempotent of a ring is an element ''a'' such that . That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that for ...
s. Clearly, the argument can be reversed and so there is a one-to-one correspondence between the direct sum decomposition into ideals and the orthogonal central idempotents summing up to the unity 1. Also, each R_i itself is a ring on its own right, the unity given by e_i, and, as a ring, ''R'' is the product ring R_1 \times \cdots \times R_n. For example, again take R = \operatorname_n(D). This ring is a simple ring; in particular, it has no nontrivial decomposition into two-sided ideals.


Types of decomposition

There are several types of direct sum decompositions that have been studied: *
Semisimple decomposition In mathematics, specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group or an algebra that is a direct sum of simple representations (also called ...
: a direct sum of simple modules. *Indecomposable decomposition: a direct sum of indecomposable modules. *A decomposition with local endomorphism rings (cf. #Azumaya's theorem): a direct sum of modules whose endomorphism rings are
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 ...
s (a ring is local if for each element ''x'', either ''x'' or 1 − ''x'' is a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
). *Serial decomposition: a direct sum of
uniserial module In abstract algebra, a uniserial module ''M'' is a module over a ring ''R'', whose submodules are totally ordered by inclusion. This means simply that for any two submodules ''N''1 and ''N''2 of ''M'', either N_1\subseteq N_2 or N_2\subseteq N_1. ...
s (a module is uniserial if the lattice of submodules is a finite chain). Since a simple module is indecomposable, a semisimple decomposition is an indecomposable decomposition (but not conversely). If the endomorphism ring of a module is local, then, in particular, it cannot have a nontrivial idempotent: the module is indecomposable. Thus, a decomposition with local endomorphism rings is an indecomposable decomposition. A direct summand is said to be ''maximal'' if it admits an indecomposable complement. A decomposition \textstyle is said to ''complement maximal direct summands'' if for each maximal direct summand ''L'' of ''M'', there exists a subset J \subset I such that :M = \left(\bigoplus_ M_j \right) \bigoplus L. Two decompositions M = \bigoplus_ M_i = \bigoplus_ N_j are said to be ''equivalent'' if there is a bijection \varphi : I \overset\to J such that for each i \in I, M_i \simeq N_. If a module admits an indecomposable decomposition complementing maximal direct summands, then any two indecomposable decompositions of the module are equivalent.


Azumaya's theorem

In the simplest form, Azumaya's theorem states: given a decomposition M = \bigoplus_ M_i such that the endomorphism ring of each M_i is
local 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 ...
(so the decomposition is indecomposable), each indecomposable decomposition of ''M'' is equivalent to this given decomposition. The more precise version of the theorem states: still given such a decomposition, if M = N \oplus K, then # if nonzero, ''N'' contains an indecomposable direct summand, # if N is indecomposable, the endomorphism ring of it is local and K is complemented by the given decomposition: #:M = M_j \oplus K and so M_j \simeq N for some j \in I, # for each i \in I, there exist direct summands N' of N and K' of K such that M = M_i \oplus N' \oplus K'. The endomorphism ring of an indecomposable module of finite length is local (e.g., by Fitting's lemma) and thus Azumaya's theorem applies to the setup of the
Krull–Schmidt theorem In mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups. Definitions We say that a group ''G ...
. Indeed, if ''M'' is a module of finite length, then, by
induction Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...
on length, it has a finite indecomposable decomposition M = \bigoplus_^n M_i, which is a decomposition with local endomorphism rings. Now, suppose we are given an indecomposable decomposition M = \bigoplus_^m N_i. Then it must be equivalent to the first one: so m = n and M_i \simeq N_ for some permutation \sigma of \. More precisely, since N_1 is indecomposable, M = M_ \bigoplus (\bigoplus_^n N_i) for some i_1. Then, since N_2 is indecomposable, M = M_ \bigoplus M_ \bigoplus (\bigoplus_^n N_i) and so on; i.e., complements to each sum \bigoplus_^n N_i can be taken to be direct sums of some M_i's. Another application is the following statement (which is a key step in the proof of
Kaplansky's theorem on projective modules In abstract algebra, Kaplansky's theorem on projective modules, first proven by Irving Kaplansky, states that a projective module over a local ring is free; where a not-necessary-commutative ring is called ''local'' if for each element ''x'', eith ...
): *Given an element x \in N, there exist a direct summand H of N and a subset J \subset I such that x \in H and H \simeq \bigoplus_ M_j. To see this, choose a finite set F \subset I such that x \in \bigoplus_ M_j. Then, writing M = N \oplus L, by Azumaya's theorem, M = (\oplus_ M_j) \oplus N_1 \oplus L_1 with some direct summands N_1, L_1 of N, L and then, by modular law, N = H \oplus N_1 with H = (\oplus_ M_j \oplus L_1) \cap N. Then, since L_1 is a direct summand of L, we can write L = L_1 \oplus L_1' and then \oplus_ M_j \simeq H \oplus L_1', which implies, since ''F'' is finite, that H \simeq \oplus_ M_j for some ''J'' by a repeated application of Azumaya's theorem. In the setup of Azumaya's theorem, if, in addition, each M_i is countably generated, then there is the following refinement (due originally to Crawley–Jónsson and later to Warfield): N is isomorphic to \bigoplus_ M_j for some subset J \subset I. (In a sense, this is an extension of Kaplansky's theorem and is proved by the two lemmas used in the proof of the theorem.) According to , it is not known whether the assumption "M_i countably generated" can be dropped; i.e., this refined version is true in general.


Decomposition of a ring

On the decomposition of a ring, the most basic but still important observation, known as the Artin–Wedderburn theorem is this: given a ring ''R'', the following are equivalent: # ''R'' is a
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 ...
; i.e., _R R is a semisimple left module. # R \simeq \prod_^r \operatorname_(D_i) where \operatorname_n(D) denotes the ring of ''n''-by-''n'' matrices and the positive
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 ...
s r, m_1, \dots, m_r are determined by ''R'' (but the D_is are not determined by ''R''). # Every left module over ''R'' is semisimple. To see the equivalence of the first two, note: if _R R \simeq \bigoplus_^r I_i^ where I_i are mutually non-isomorphic left minimal ideals, then, with the view that endomorphisms act from the right, :R \simeq \operatorname(_R R) \simeq \bigoplus_^r \operatorname(I_i^) where each \operatorname(I_i^) can be viewed as the matrix ring over the division ring D_i = \operatorname(I_i). (The converse is because the decomposition of 2. is equivalent to a decomposition into minimal left ideals = simple left submodules.) The equivalence 1. \Leftrightarrow 3. is because every module is a
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of a free module and a quotient of a semisimple module is clearly semisimple.


See also

* Pure-injective module


Notes


References

* * Frank W. Anderson
Lectures on Non-Commutative Rings
University of Oregon, Fall, 2002. * * * Y. Lam, Bass’s work in ring theory and projective modules R 1732042* * R. Warfield: Exchange rings and decompositions of modules, Math. Annalen 199(1972), 31-36. {{algebra-stub Module theory