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, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, a decomposition of a module is a way to write a module 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, m ...
. 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 itself ...
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 ...
s. Given a
ring (The) Ring(s) 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 Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...
, 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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
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. Indecomposable is a weaker notion than simple module (which is also sometimes called irreducible module): simple ...
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 (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since t ...
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 Arts, entertainment, and media * ''Local'' (comics), a limited series comic book by Bria ...
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 ...
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. 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) of
matrix ring In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication. The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'') (alternat ...
s over
division ring In algebra, a division ring, also called a skew field (or, occasionally, a sfield), 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 multiplicativ ...
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 ...
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, unc ...
. Indeed, if M = \bigoplus_ M_i, then, for each i \in I, the
linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...
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 g ...
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 or picture is a visual representation. An image can be two-dimensional, such as a drawing, painting, or photograph, or three-dimensional, such as a carving or sculpture. Images may be displayed through other media, including a project ...
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 (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the ...
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 mathematics, an idempotent element or simply idempotent of a ring (mathematics), ring is an element such that . That is, the element is idempotent under the ring's multiplication. Mathematical induction, Inductively the ...
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 mathematics, more specifically in 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 algebraic varieties or manifolds, or of ...
s (a ring is local if for each element ''x'', either ''x'' or 1 − ''x'' is a
unit Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...
). *Serial decomposition: a direct sum of
uniserial module In abstract algebra, a uniserial module ''M'' is a module (mathematics), module over a ring (mathematics), ring ''R'', whose submodules are total order, totally ordered by inclusion (set theory), inclusion. This means simply that for any two submo ...
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 In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
\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 Arts, entertainment, and media * ''Local'' (comics), a limited series comic book by Bria ...
(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 Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...
is local (e.g., by
Fitting's lemma In mathematics, the Fitting lemma – named after the mathematician Hans Fitting – is a basic statement in abstract algebra. Suppose ''M'' is a module over some ring. If ''M'' is indecomposable and has finite length, then every endomorphism of ...
) and thus Azumaya's theorem applies to the setup of the Krull–Schmidt theorem. Indeed, if ''M'' is a module of finite length, then, by induction 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 In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...
\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 Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...
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-necessarily-commutative ring is called ''local'' if for each element ''x'', eit ...
): *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 In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition, ;Modular law: implies where are arbitrary elements in the lattice,  ≤  is the partial order, and &nb ...
, 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 In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
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 Wedderburn-Artin 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 itself ...
; i.e., _R R is a semisimple left module. # R \cong \prod_^r \operatorname_(D_i) for
division ring In algebra, a division ring, also called a skew field (or, occasionally, a sfield), 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 multiplicativ ...
s D_1, \dots, D_r, where \operatorname_n(D_i) denotes the ring of ''n''-by-''n'' matrices with entries in D_i, and the positive
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s r, the division rings D_1, \dots , D_r, and the positive integers m_1, \dots, m_r are determined (the latter two up to permutation) by ''R'' # Every left module over ''R'' is semisimple. To show 1. \Rightarrow 2., first note that if R is semisimple then we have an isomorphism of left R-modules _R R \cong \bigoplus_^r I_i^ where I_i are mutually non-isomorphic minimal left ideals. Then, with the view that endomorphisms act from the right, :R \cong \operatorname(_R R) \cong \bigoplus_^r \operatorname(I_i^) where each \operatorname(I_i^) can be viewed as the matrix ring over D_i = \operatorname(I_i), which is a division ring by
Schur's Lemma In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations of a gro ...
. The converse holds because the decomposition of 2. is equivalent to a decomposition into minimal left ideals = simple left submodules. The equivalence 1. \Leftrightarrow 3. holds because every module is a
quotient In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
of a
free module In mathematics, a free module is a module that has a ''basis'', that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commu ...
, and a quotient of a semisimple module is 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* {{cite book , last1=Procesi , first1=Claudio , author-link=Claudio Procesi , title=Lie groups : an approach through invariants and representations , date=2007 , publisher=Springer , location=New York , isbn=9780387260402 * R. Warfield: Exchange rings and decompositions of modules, Math. Annalen 199(1972), 31–36. Module theory