decomposition of a module

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

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.{{harvnb|Anderson|Fuller|1992|loc=Corollary 6.19. and Corollary 6.20.}} Indeed, if $M = \bigoplus_{i \in I} M_i$ , then, for each $i \in I$ , the linear endomorphism $e_i : M \to M_i \hookrightarrow M$ given by the natural projection followed by the natural inclusion is an idempotent. They are clearly orthogonal to each other ( $e_i e_j = 0$ for $i \ne j$ ) and they sum up to the identity map:

: $1_{\operatorname{M}} = \sum_{i \in I} e_i$

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 $\{ e_i \}_{i \in I}$ such that only finitely many $e_i(x)$ are nonzero for each $x \in M$ and $1_R = \sum_{a \in A} e_a \in \bigoplus_{a \in A} I_a$ , 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 ${}_R R$ 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.{{harvnb|Procesi|2007|loc=Ch.6., § 1.3.}}

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 $\textstyle{1 = \sum_1^n e_i}$ . 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; i.e., they are central idempotents.{{harvnb|Anderson|Fuller|1992|loc=Proposion 7.6.}} 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 $R = \operatorname{M}_n(D)$ . 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:

Semisimple decomposition: a direct sum of simple modules.
Indecomposable decomposition: a direct sum of indecomposable modules.
A decomposition with local endomorphism rings{{harv|Jacobson|2009|loc=A paragraph before Theorem 3.6.}} calls a module strongly indecomposable if nonzero and has local endomorphism ring. (cf. #Azumaya's theorem): a direct sum of modules whose endomorphism rings are local rings (a ring is local if for each element x, either x or 1 − x is a unit).
Serial decomposition: a direct sum of uniserial modules (a module is uniserial if the lattice of submodules is a finite chain{{harvnb|Anderson|Fuller|1992|loc=§ 32.}}).

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 $\textstyle{M = \bigoplus_{i \in I} M_i}$ 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

: $M = \left(\bigoplus_{j \in J} M_j \right) \bigoplus L.$ {{harvnb|Anderson|Fuller|1992|loc=§ 12.}}

Two decompositions $M = \bigoplus_{i \in I} M_i = \bigoplus_{j \in J} N_j$ are said to be equivalent if there is a bijection $\varphi : I \overset{\sim}\to J$ such that for each $i \in I$ , $M_i \simeq N_{\varphi(i)}$ . If a module admits an indecomposable decomposition complementing maximal direct summands, then any two indecomposable decompositions of the module are equivalent.{{harvnb|Anderson|Fuller|1992|loc=Theorrm 12.4.}}

Azumaya's theorem

In the simplest form, Azumaya's theorem states:{{harvnb|Facchini|1998|loc=Theorem 2.12.}} given a decomposition $M = \bigoplus_{i \in I} M_i$ such that the endomorphism ring of each $M_i$ 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:{{harvnb|Anderson|Fuller|1992|loc=Theorem 12.6. and Lemma 26.4.}} 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{{harvnb|Facchini|1998|loc=Lemma 2.11.}} 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. Indeed, if M is a module of finite length, then, by induction on length, it has a finite indecomposable decomposition $M = \bigoplus_{i=1}^n M_i$ , which is a decomposition with local endomorphism rings. Now, suppose we are given an indecomposable decomposition $M = \bigoplus_{i=1}^m N_i$ . Then it must be equivalent to the first one: so $m = n$ and $M_i \simeq N_{\sigma(i)}$ for some permutation $\sigma$ of $\{ 1, \dots, n \}$ . More precisely, since $N_1$ is indecomposable, $M = M_{i_1} \bigoplus (\bigoplus_{i=2}^n N_i)$ for some $i_1$ . Then, since $N_2$ is indecomposable, $M = M_{i_1} \bigoplus M_{i_2} \bigoplus (\bigoplus_{i=3}^n N_i)$ and so on; i.e., complements to each sum $\bigoplus_{i=l}^n N_i$ 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):

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 $H \simeq \bigoplus_{j \in J} M_j$ .

To see this, choose a finite set $F \subset I$ such that $x \in \bigoplus_{j \in F} M_j$ . Then, writing $M = N \oplus L$ , by Azumaya's theorem, $M = (\oplus_{j \in F} M_j) \oplus N_1 \oplus L_1$ with some direct summands $N_1, L_1$ of $N, L$ and then, by modular law, $N = H \oplus N_1$ with $H = (\oplus_{j \in F} M_j \oplus L_1) \cap N$ . Then, since $L_1$ is a direct summand of $L$ , we can write $L = L_1 \oplus L_1'$ and then $\oplus_{j \in F} M_j \simeq H \oplus L_1'$ , which implies, since F is finite, that $H \simeq \oplus_{j \in J} M_j$ 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 $\bigoplus_{j \in J} M_j$ for some subset $J \subset I$ .{{harvnb|Facchini|1998|loc=Corollary 2.55.}} (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 {{harv|Facchini|1998}}, 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; i.e., ${}_R R$ is a semisimple left module.
$R \cong \prod_{i=1}^r \operatorname{M}_{m_i}(D_i)$ for division rings $D_1, \dots, D_r$ , where $\operatorname{M}_n(D_i)$ denotes the ring of n-by-n matrices with entries in $D_i$ , and the positive integers $r$ , 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 ${}_R R \cong \bigoplus_{i=1}^r I_i^{\oplus m_i}$ where $I_i$ are mutually non-isomorphic minimal left ideals. Then, with the view that endomorphisms act from the right,

: $R \cong \operatorname{End}({}_R R) \cong \bigoplus_{i=1}^r \operatorname{End}(I_i^{\oplus m_i})$

where each $\operatorname{End}(I_i^{\oplus m_i})$ can be viewed as the matrix ring over $D_i = \operatorname{End}(I_i)$ , 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.

Notes

References

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Category:Module theory