CHAPTER 1
The Krull-Remak-Schmidt Theorem
In this chapter we will prove the Krull-Remak-Schmidt uniqueness
theorem for direct-sum decompositions of finitely generated modules
over complete local rings. The first such theorem, in the context of
finite groups, was stated by Wedderburn [Wed09]: Let G be a finite
group with two direct-product decompositions G = H1 × · · · × Hm and
G = K1 × · · · × Kn, where each Hi and each Kj is indecomposable.
Then m = n, and, after renumbering, Hi
∼
= Ki for each i. In 1911
Remak [Rem11] gave a complete proof, and actually proved more:
Hi and Ki are centrally isomorphic, that is, there are isomorphisms
fi : Hi −→ Ki such that
x−1f(x)
is in the center of G for each x ∈ Hi,
i = 1,...,m. These results were extended to groups with operators sat-
isfying the ascending and descending chain conditions by Krull [Kru25]
and Schmidt [Sch29]. In 1950 Azumaya [Azu50] proved an analogous
result for possibly infinite direct sums of modules, with the assump-
tion that the endomorphism ring of each factor is local in the non-
commutative sense.
§1. KRS in an additive category
Looking ahead to an application in Chapter 3, we will clutter things
up slightly by working in an additive category, rather than a category
of modules. An additive category is a category A with 0-object such
that (i) HomA(M1,M2) is an abelian group for each pair M1, M2 of
objects, (ii) composition is bilinear, and (iii) every finite set of objects
has a biproduct. A biproduct of M1,... , Mm consists of an object M
together with maps ui : Mi −→ M and pi : M −→ Mi, i = 1,...,m,
such that piuj = δij and u1p1 + · · · + umpm = 1M . We denote the
biproduct by M1 ⊕ · · · ⊕ Mm.
We will need an additional condition on our additive category, that
idempotents split (cf. [Bas68, Chapter I, §3, p. 19]). Given an object
M and an idempotent e ∈ EndA(M), we say that e splits provided there
is a factorization M
p
− − → K
u
−→ − M such that e = up and pu = 1K.
The reader is probably familiar with the notion of an abelian cat-
egory, that is, an additive category in which every map has a kernel
1
http://dx.doi.org/10.1090/surv/181/01
