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. 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 determine a direct sum decomposition by taking $$M_i$$ to be the images 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
 * $${}_R R = \bigoplus_{a \in A} I_a$$

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 $$\operatorname{End}({}_R R) \simeq R$$. The summation of idempotent endomorphisms corresponds to the decomposition of the unity of R: $1_R = \sum_{a \in A} e_a \in \bigoplus_{a \in A} I_a$, 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.

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. 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 (cf. ): 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 ).

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
 * $$M = \left(\bigoplus_{j \in J} M_j \right) \bigoplus L.$$

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.

Azumaya's theorem
In the simplest form, Azumaya's theorem states: 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: still given such a decomposition, if $$M = N \oplus K$$, 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 $$M_j \simeq N$$ for some $$j \in I$$,
 * 1) 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): 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.
 * 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 $H \simeq \bigoplus_{j \in J} M_j$.

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 $$J \subset I$$. (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:
 * 1) R is a semisimple ring; i.e., $${}_R R$$ is a semisimple left module.
 * 2) $$R \cong \prod_{i=1}^r \operatorname{M}_{m_i}(D_i)$$ 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
 * 3) 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.