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

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. ^[1] Indeed, if ${\textstyle M=\bigoplus _{i\in I}M_{i}}$, then, for each ${\displaystyle i\in I}$, the linear endomorphism ${\displaystyle e_{i}:M\to M_{i}\hookrightarrow M}$ given by the natural projection followed by the natural inclusion is an idempotent. They are clearly orthogonal to each other (${\displaystyle e_{i}e_{j}=0}$ for ${\displaystyle i\neq j}$) and they sum up to the identity map:

${\displaystyle 1_{\operatorname {M} }=\sum _{i\in I}e_{i}}$

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 ${\displaystyle \{e_{i}\}_{i\in I}}$ such that only finitely many ${\displaystyle e_{i}(x)}$ are nonzero for each ${\displaystyle x\in M}$ and $\sum e_{i}=1_{M}$ determine a direct sum decomposition by taking ${\displaystyle M_{i}}$ to be the images of ${\displaystyle e_{i}}$.

This fact already puts some constraints on a possible decomposition of a ring: given a ring ${\displaystyle R}$, suppose there is a decomposition

${\displaystyle {}_{R}R=\bigoplus _{a\in A}I_{a}}$

of ${\displaystyle R}$ as a left module over itself, where ${\displaystyle I_{a}}$ are left submodules; i.e., left ideals. Each endomorphism ${\displaystyle {}_{R}R\to {}_{R}R}$ can be identified with a right multiplication by an element of R; thus, ${\displaystyle I_{a}=Re_{a}}$ where ${\displaystyle e_{a}}$ are idempotents of ${\displaystyle \operatorname {End} ({}_{R}R)\simeq R}$. ^[2] The summation of idempotent endomorphisms corresponds to the decomposition of the unity of R: ${\textstyle 1_{R}=\sum _{a\in A}e_{a}\in \bigoplus _{a\in A}I_{a}}$, which is necessarily a finite sum; in particular, ${\displaystyle A}$ must be a finite set.

For example, take ${\displaystyle R=\operatorname {M} _{n}(D)}$, the ring of n-by-n matrices over a division ring D. Then ${\displaystyle {}_{R}R}$ is the direct sum of n copies of ${\displaystyle D^{n}}$, the columns; each column is a simple left R-submodule or, in other words, a minimal left ideal. ^[3]

Let R be a ring. Suppose there is a (necessarily finite) decomposition of it as a left module over itself

${\displaystyle {}_{R}R=R_{1}\oplus \cdots \oplus R_{n}}$

into two-sided ideals${\displaystyle R_{i}}$ of R. As above, ${\displaystyle R_{i}=Re_{i}}$ for some orthogonal idempotents ${\displaystyle e_{i}}$ such that ${\displaystyle \textstyle {1=\sum _{1}^{n}e_{i}}}$. Since ${\displaystyle R_{i}}$ is an ideal, ${\displaystyle e_{i}R\subset R_{i}}$ and so ${\displaystyle e_{i}Re_{j}\subset R_{i}\cap R_{j}=0}$ for ${\displaystyle i\neq j}$. Then, for each i,

${\displaystyle e_{i}r=\sum _{j}e_{j}re_{i}=\sum _{j}e_{i}re_{j}=re_{i}.}$

That is, the ${\displaystyle e_{i}}$ are in the center; i.e., they are central idempotents. ^[4] 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 ${\displaystyle R_{i}}$ itself is a ring on its own right, the unity given by ${\displaystyle e_{i}}$, and, as a ring, R is the product ring ${\displaystyle R_{1}\times \cdots \times R_{n}.}$

For example, again take ${\displaystyle R=\operatorname {M} _{n}(D)}$. 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 ^[5] (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 ^[6] ).

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 ${\displaystyle \textstyle {M=\bigoplus _{i\in I}M_{i}}}$ is said to complement maximal direct summands if for each maximal direct summand L of M, there exists a subset ${\displaystyle J\subset I}$ such that

${\displaystyle M=\left(\bigoplus _{j\in J}M_{j}\right)\bigoplus L.}$ ^[7]

Two decompositions ${\displaystyle M=\bigoplus _{i\in I}M_{i}=\bigoplus _{j\in J}N_{j}}$ are said to be equivalent if there is a bijection ${\displaystyle \varphi :I{\overset {\sim }{\to }}J}$ such that for each ${\displaystyle i\in I}$, ${\displaystyle M_{i}\simeq N_{\varphi (i)}}$. ^[7] If a module admits an indecomposable decomposition complementing maximal direct summands, then any two indecomposable decompositions of the module are equivalent. ^[8]

Azumaya's theorem

In the simplest form, Azumaya's theorem states: ^[9] given a decomposition ${\displaystyle M=\bigoplus _{i\in I}M_{i}}$ such that the endomorphism ring of each ${\displaystyle M_{i}}$ 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: ^[10] still given such a decomposition, if ${\displaystyle M=N\oplus K}$, then

1. if nonzero, N contains an indecomposable direct summand,
2. if ${\displaystyle N}$ is indecomposable, the endomorphism ring of it is local ^[11] and ${\displaystyle K}$ is complemented by the given decomposition:

   ${\textstyle M=M_{j}\oplus K}$ and so ${\displaystyle M_{j}\simeq N}$ for some ${\displaystyle j\in I}$,

3. for each ${\displaystyle i\in I}$, there exist direct summands ${\displaystyle N'}$ of ${\displaystyle N}$ and ${\displaystyle K'}$ of ${\displaystyle K}$ such that ${\displaystyle M=M_{i}\oplus N'\oplus K'}$.

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 ${\textstyle M=\bigoplus _{i=1}^{n}M_{i}}$, which is a decomposition with local endomorphism rings. Now, suppose we are given an indecomposable decomposition ${\textstyle M=\bigoplus _{i=1}^{m}N_{i}}$. Then it must be equivalent to the first one: so ${\displaystyle m=n}$ and ${\displaystyle M_{i}\simeq N_{\sigma (i)}}$ for some permutation ${\displaystyle \sigma }$ of ${\displaystyle \{1,\dots ,n\}}$. More precisely, since ${\displaystyle N_{1}}$ is indecomposable, ${\textstyle M=M_{i_{1}}\bigoplus (\bigoplus _{i=2}^{n}N_{i})}$ for some ${\displaystyle i_{1}}$. Then, since ${\displaystyle N_{2}}$ is indecomposable, ${\textstyle M=M_{i_{1}}\bigoplus M_{i_{2}}\bigoplus (\bigoplus _{i=3}^{n}N_{i})}$ and so on; i.e., complements to each sum ${\textstyle \bigoplus _{i=l}^{n}N_{i}}$ can be taken to be direct sums of some ${\displaystyle M_{i}}$'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 ${\displaystyle x\in N}$, there exist a direct summand ${\displaystyle H}$ of ${\displaystyle N}$ and a subset ${\displaystyle J\subset I}$ such that ${\displaystyle x\in H}$ and ${\textstyle H\simeq \bigoplus _{j\in J}M_{j}}$.

To see this, choose a finite set ${\displaystyle F\subset I}$ such that ${\textstyle x\in \bigoplus _{j\in F}M_{j}}$. Then, writing ${\displaystyle M=N\oplus L}$, by Azumaya's theorem, ${\displaystyle M=(\oplus _{j\in F}M_{j})\oplus N_{1}\oplus L_{1}}$ with some direct summands ${\displaystyle N_{1},L_{1}}$ of ${\displaystyle N,L}$ and then, by modular law, ${\displaystyle N=H\oplus N_{1}}$ with ${\displaystyle H=(\oplus _{j\in F}M_{j}\oplus L_{1})\cap N}$. Then, since ${\displaystyle L_{1}}$ is a direct summand of ${\displaystyle L}$, we can write ${\displaystyle L=L_{1}\oplus L_{1}'}$ and then ${\displaystyle \oplus _{j\in F}M_{j}\simeq H\oplus L_{1}'}$, which implies, since F is finite, that ${\displaystyle H\simeq \oplus _{j\in J}M_{j}}$ for some J by a repeated application of Azumaya's theorem.

In the setup of Azumaya's theorem, if, in addition, each ${\displaystyle M_{i}}$ is countably generated, then there is the following refinement (due originally to Crawley–Jónsson and later to Warfield): ${\displaystyle N}$ is isomorphic to ${\displaystyle \bigoplus _{j\in J}M_{j}}$ for some subset ${\displaystyle J\subset I}$. ^[12] (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 ( Facchini 1998 ), it is not known whether the assumption "${\displaystyle M_{i}}$ 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., ${\displaystyle {}_{R}R}$ is a semisimple left module.
2. ${\displaystyle R\cong \prod _{i=1}^{r}\operatorname {M} _{m_{i}}(D_{i})}$ for division rings ${\displaystyle D_{1},\dots ,D_{r}}$, where ${\displaystyle \operatorname {M} _{n}(D_{i})}$ denotes the ring of n-by-n matrices with entries in ${\displaystyle D_{i}}$, and the positive integers ${\displaystyle r}$, the division rings ${\displaystyle D_{1},\dots ,D_{r}}$, and the positive integers ${\displaystyle m_{1},\dots ,m_{r}}$ are determined (the latter two up to permutation) by R
3. Every left module over R is semisimple.

To show 1. ${\displaystyle \Rightarrow }$ 2., first note that if ${\displaystyle R}$ is semisimple then we have an isomorphism of left ${\displaystyle R}$-modules ${\textstyle {}_{R}R\cong \bigoplus _{i=1}^{r}I_{i}^{\oplus m_{i}}}$ where ${\displaystyle I_{i}}$ are mutually non-isomorphic minimal left ideals. Then, with the view that endomorphisms act from the right,

${\displaystyle R\cong \operatorname {End} ({}_{R}R)\cong \bigoplus _{i=1}^{r}\operatorname {End} (I_{i}^{\oplus m_{i}})}$

where each ${\displaystyle \operatorname {End} (I_{i}^{\oplus m_{i}})}$ can be viewed as the matrix ring over ${\displaystyle D_{i}=\operatorname {End} (I_{i})}$, 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. ${\displaystyle \Leftrightarrow }$ 3. holds because every module is a quotient of a free module, and a quotient of a semisimple module is semisimple.

See also

• Pure-injective module

Notes

1. Anderson & Fuller 1992 , Corollary 6.19. and Corollary 6.20.
2. 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.
3. Procesi 2007 , Ch.6., § 1.3.
4. Anderson & Fuller 1992 , Proposion 7.6.
5. ( Jacobson 2009 , A paragraph before Theorem 3.6.) calls a module strongly indecomposable if nonzero and has local endomorphism ring.
6. Anderson & Fuller 1992 , § 32.
7. Anderson & Fuller 1992 , § 12.
8. Anderson & Fuller 1992 , Theorrm 12.4.
9. Facchini 1998 , Theorem 2.12.
10. Anderson & Fuller 1992 , Theorem 12.6. and Lemma 26.4.
11. Facchini 1998 , Lemma 2.11.
12. Facchini 1998 , Corollary 2.55.

References

• Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, Graduate Texts in Mathematics, vol. 13 (2 ed.), New York: Springer-Verlag, pp. x+376, doi:10.1007/978-1-4612-4418-9, ISBN   0-387-97845-3, MR   1245487
• Frank W. Anderson, Lectures on Non-Commutative Rings Archived 2021-06-13 at the Wayback Machine , University of Oregon, Fall, 2002.
• Facchini, Alberto (16 June 1998). Module Theory: Endomorphism rings and direct sum decompositions in some classes of modules. Springer Science & Business Media. ISBN   978-3-7643-5908-9.
• Jacobson, Nathan (2009), Basic algebra, vol. 2 (2nd ed.), Dover, ISBN   978-0-486-47187-7
• Y. Lam, Bass's work in ring theory and projective modules [MR 1732042]
• Procesi, Claudio (2007). Lie groups : an approach through invariants and representations. New York: Springer. ISBN   9780387260402.
• R. Warfield: Exchange rings and decompositions of modules, Math. Annalen 199(1972), 31–36.