Decomposition of a module explained

In abstract algebra, a decomposition of a module is a way to write a module as a direct sum of modules. A type of a decomposition is often used to define or characterize modules: for example, a semisimple module is a module that has a decomposition into simple modules. Given a ring, 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 every module over it is a semisimple module.

An indecomposable module is a module that is not a direct sum of two nonzero submodules. Azumaya's theorem states that if a module has an decomposition into modules with local endomorphism rings, then all decompositions into indecomposable modules are equivalent to each other; a special case of this, especially in group theory, 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 rings over division rings (this observation is known as the Artin–Wedderburn theorem).

Idempotents and decompositions

See main article: Idempotent element.

To give a direct sum decomposition of a module into submodules is the same as to give orthogonal idempotents in the endomorphism ring of the module that sum up to the identity map. Indeed, if M = \bigoplus_ M_i, then, for each

i\inI

, the linear endomorphism

ei:M\toMi\hookrightarrowM

given by the natural projection followed by the natural inclusion is an idempotent. They are clearly orthogonal to each other (

eiej=0

for

i\nej

) and they sum up to the identity map:

1\operatorname{M

} = \sum_ e_ias 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

\{ei\}i

such that only finitely many

ei(x)

are nonzero for each

x\inM

and

Mi

to be the images of

ei

.

This fact already puts some constraints on a possible decomposition of a ring: given a ring

R

, suppose there is a decomposition

{}RR=oplusaIa

of

R

as a left module over itself, where

Ia

are left submodules; i.e., left ideals. Each endomorphism

{}RR\to{}RR

can be identified with a right multiplication by an element of R; thus,

Ia=Rea

where

ea

are idempotents of

\operatorname{End}({}RR)\simeqR

.[1] 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{M}n(D)

, the ring of n-by-n matrices over a division ring D. Then

{}RR

is the direct sum of n copies of

Dn

, 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

{}RR=R1 ⊕ … ⊕ Rn

into two-sided ideals

Ri

of R. As above,

Ri=Rei

for some orthogonal idempotents

ei

such that

style{1=

n
\sum
1

ei}

. Since

Ri

is an ideal,

eiR\subsetRi

and so

eiRej\subsetRi\capRj=0

for

i\nej

. Then, for each i,

eir=\sumjejrei=\sumjeirej=rei.

That is, the

ei

are in the center; i.e., they are central idempotents. 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

Ri

itself is a ring on its own right, the unity given by

ei

, and, as a ring, R is the product ring

R1 x … x Rn.

For example, again take

R=\operatorname{M}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:

a direct sum of simple modules.

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

style{M=oplusiMi}

is said to complement maximal direct summands if for each maximal direct summand L of M, there exists a subset

J\subsetI

such that

M=\left(oplusjMj\right)oplusL.

Two decompositions

M=oplusiMi=oplusjNj

are said to be equivalent if there is a bijection

\varphi:I\overset{\sim}\toJ

such that for each

i\inI

,

Mi\simeqN\varphi(i)

. 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=oplusiMi

such that the endomorphism ring of each

Mi

is local (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 ⊕ K

, then
  1. if nonzero, N contains an indecomposable direct summand,
  2. 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

Mj\simeqN

for some

j\inI

,
  1. for each

i\inI

, there exist direct summands

N'

of

N

and

K'

of

K

such that

M=Mi ⊕ N' ⊕ 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. 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

Mi\simeqN\sigma(i)

for some permutation

\sigma

of

\{1,...,n\}

. More precisely, since

N1

is indecomposable, M = M_ \bigoplus (\bigoplus_^n N_i) for some

i1

. Then, since

N2

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

Mi

's.

Another application is the following statement (which is a key step in the proof of Kaplansky's theorem on projective modules):

x\inN

, there exist a direct summand

H

of

N

and a subset

J\subsetI

such that

x\inH

and H \simeq \bigoplus_ M_j.To see this, choose a finite set

F\subsetI

such that x \in \bigoplus_ M_j. Then, writing

M=N ⊕ L

, by Azumaya's theorem,

M=( ⊕ jMj) ⊕ N1 ⊕ L1

with some direct summands

N1,L1

of

N,L

and then, by modular law,

N=H ⊕ N1

with

H=( ⊕ jMj ⊕ L1)\capN

. Then, since

L1

is a direct summand of

L

, we can write

L=L1 ⊕ L1'

and then

⊕ jMj\simeqH ⊕ L1'

, which implies, since F is finite, that

H\simeq ⊕ jMj

for some J by a repeated application of Azumaya's theorem.

In the setup of Azumaya's theorem, if, in addition, each

Mi

is countably generated, then there is the following refinement (due originally to Crawley–Jónsson and later to Warfield):

N

is isomorphic to

oplusjMj

for some subset

J\subsetI

. (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 "

Mi

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:

  1. R is a semisimple ring; i.e.,

{}RR

is a semisimple left module.

R\cong

r
\prod
i=1
\operatorname{M}
mi

(Di)

for division rings

D1,...,Dr

, where

\operatorname{M}n(Di)

denotes the ring of n-by-n matrices with entries in

Di

, and the positive integers

r

, the division rings

D1,...,Dr

, and the positive integers

m1,...,mr

are determined (the latter two up to permutation) by R
  1. Every left module over R is semisimple.

To show 1.

⇒

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

Ii

are mutually non-isomorphic minimal left ideals. Then, with the view that endomorphisms act from the right,

R\cong\operatorname{End}({}RR)\cong

r
oplus
i=1
⊕ mi
\operatorname{End}(I
i

)

where each
⊕ mi
\operatorname{End}(I
i

)

can be viewed as the matrix ring over

Di=\operatorname{End}(Ii)

, which is a division ring by Schur's Lemma. 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 of a free module, and a quotient of a semisimple module is semisimple.

See also

References

Notes and References

  1. Here, the endomorphism ring is thought of as acting from the right; if it acts from the left, this identification is for the opposite ring of R.
  2. calls a module strongly indecomposable if nonzero and has local endomorphism ring.

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