ASASHIBA, HIDETO

SUMIOKA, TAKESHI

Asashiba, Hideto

Sumioka, Takeshi

Osaka University Knowledge Archive

On Krull-Schmidt's theorem and the indecomposability of amalgamated sum

Osaka City University Repository

Asashiba, H. and Sumioka, T.
Osaka J. Math.
20 (1983), 321-329
ON KRULL-SCHMIDT'S THEOREM AND
THE INDECOMPOSABILITY OF
AMALGAMATED SUM
HIDETO ASASHIBA AND TAKESHI SUMIOKA
(Received June 18, 1981)
Let R be a ring and M an i?-module. If M has direct decompositions
M=®Li=(BMj into completely indecomposable i?-modules L, and M. .
Then, by Krull-Remak-Schmidt-Azumaya's theorem ([1]), it holds that for
any finite subset ]' of /, there exists a finite subset /' of / such that
(1) M= 0 L t θ ( θ MA. Then however it is not necessarily satisfied
ί€=l' j(ΞJ-j'
(2) M= βMj®( 0 L^ In § 1, we shall show that (1) and (2) hold simul-
taneously for suitable subset /' of /. The assertion is first showed in case M
is semi-simple (in any completely reducible Grothendieck category). Then it
is valid in general case, using the method of Harada and Sai [3, Corollary 1,
p. 334]. But when the index set / is finite, we give an elementary proof for it.
Next, we consider finitely generated indecomposable modules over right
artinian rings. In [4] and [5], Tachikawa investigated algebras of right local
type (i.e. every indecomposable right module has the simple top) and of local
or colocal type. To prove his assertions, he constructed indecomposable
modules which were obtained by amalgamated sums (they were called inter-
lacings there). In § 2, we shall slightly generalize his method, and give suffi-
cient conditions for amalgamated sums to be indecomposable.
The authors wish to express their appreciation to Professor M. Harada
and Mr. T. Inoue. The former suggested them that Theorem 1.3 (Theorem
1.7) is obtained from Lemma 1.1 (Lemma 1.6) using the method in [3], and
the latter simplified their proof of Lemma 1.1 by his own method. As its proof,
we take his own.
Throughout this note, R denotes a ring with unity and i?-modules are
(unital) right i?-modules unless otherwise stated. For i?-modules £,(/£/),
we use a notation 0/L, instead of φL, . If/: L-^M is a homomorphism and
U is a submodule of L, we also denote the restriction map to L' of/: L->M
by /: L'->M. Let / be a set and Ii a subset of / for each /=!,•••, n.
322 H . ASASHIBA AND T . SUMIOKA
If 1=1x1) * 'UI
Λ
 and Ijf]Ik=φ for ally and k(j^k)y we say that the union
Λ U — U /„ is a partition (of /), and denote it by / = / i Π —ΠΛ
1. Krull-Remak-Schmidt-Azumaya's theorem
In this section, we study a generalization of Krull-Remak-Schmidt-Azu-
maya's theorem. The following lemma is basic for our results.
Lemma 1.1. Let M=L1®-~(BLn be a simple decomposition of a semi-
simple R-module M. Then for any direct decomposition M=M1®M2y there
exists a partition {1, •••, n}=I
x
YlI2 such that
(fT\ T \/T\ Ά/T Ά/f ST\( (Ύ\ T \
— \vD'/1 -
ί
-
/
ί/vχ/-ί'-ί2 — -^  "-* 1 vL/y vΓ/Λ> i) '
Proof. We prove the assertion by induction on n. Let L[ and L['
denote the images of L
λ
 under the projections M-^M
λ
 and M->M2y respec-
tively. Then it holds either M==L{®L2®•••©£„ or M=L['(&L2(B — ®Ln.
We may assume M=Lί©L 2 © ©Ln. Under the canonical homomorphism
M-^M/Lί, we denote the image of Nby N for every submodule N of M. Then
M=ί/ 2 © φLn=ikf1φikί2. By inductional assumption, there exists a parti-
tion {2, '••yn}=I{U.I'2 such that M=(®^Li)φM2^M1φ(I^Li). Hence we
have M=L{-{-((BIi'Li)-\- M2 = Mx + ί^Bj^Li). Comparing the composition
lengths of the three terms, we see the sums are direct. Then it follows M=
L
ι
@{@Ii,Li)Q)M2 and hence our assertion is satisfied for /2=:{l}U/ί and
From Lemma 1.1, we can obtain Theorem 1.3 using the argument in the
proof of Harada and Sai [3, Corollary 1, p. 334]. We shall however give an
elementary proof of it. (See [3, Lemma 2] for the following Lemma 1.2 and
Remark.)
Lemma 1.2. Let e and f be idempotents of a ring R. If R=eRQ)fR, then
we have R=eR®fRy where R=^RjJy J the Jacobson radical of R and e=e-\-Jyf=f+J<=RIJ.
Proof. Assume R=eR®fR. Then R=eR+fR since JR is small in RR.
Hence (l—f)R=(l—f)eRy which implies that a left multiplication map eR->
(1—f)R (ea\-^(l—f)ea) by (1—/) is an epimorphism. Since (1—f)R is projec-
tive, we have a split exact sequence
0->eRΓifR->eR-> (l-f)R -> 0 .
By the assumption, however, eRf] fRcieJ and hence eRf]fR is small in eR.
Therefore eRf)fR=0 and so R=eR®fR.
REMARK. L e m m a 1.2 holds m o r e generally. L e t ely ~yen be idempotents
KRULL-SCHMIDT'S THEOREM 323
of R. Using the notation of Lemma 1.2, if R=e
λ
R®~@e
n
R, then R=
n
^jRS φ^i? . In fact, put P, =£,-/? and consider an external direct sum ©P, .
Then we have a commutative diagram
I"
•* σ
R > R
—
 n
 —
RRwith canonical maps. Since σ and T are the projective covers of =
p is an isomorphism. This shows R=e
ι
R® "@e
n
R.
Recall an jR-module M is completely indecomposable provided its endmor-
phism ring ΈndR(M) is a local ring. A direct decomposition M=^ILi is
called a completely indecomposable decomposition if L, is completely indecompo-
sable for each
Theorem 1.3. Z>£ M—Li©---©!^ fo tf completely indecomposable decom-
position of an R-module M. Then for any direct decomposition M=M1φM2
there exists a partition {1, •••, w}=/iΠ^2 m c n that
M = ( 0 7 i Li)®M2 =
Proof. Let e{ denote the composition map of a projection M->L{ and
an injection Li-*M9 and £, also the composition map M->Mj->M of canonical
maps. Put S=EndR(M) and S=SIJ(S), and denote x+J(S)<=SIJ(S) by £
for tfeS, where J(S) is the Jacobson radical of S. Then, since S is a semi-
perfect ring and el9 •••,£„ are mutually orthogonal primitive idempotents, and
£i> gz a r e orthogonal idempotents, we have S=e1S(B-~(BenS=g1S®g2S and
each βf S is a simple S-module. Therefore by Lemma 1.1 there exists a parti-
tion {1, —,n}= /
x
 Π Λ such that S=f1S®g2S=g1S®f2Sy where /1 = 2 /^..
y 2 = 2 / 2 ^.. Hence by Lemma 1.2, S=f1S®g2S=g1S®f2S. Then it is easy
to see that M = ( 0 / i L f ) 0 M 2 = M 1 0 ( 0 / 2 L f . ) .
Let M = M ! 0 M 2 be a decomposition and let ^ : M-*./^ denote the pro-
jection. For a submodule L of M the restriction map ^ r L->M
λ
 is iscmor-
phic if and only if M—LQ)M2. Therefore putting Theorem 1.3 in this way,
we have
n
Lemma 1.4 Let L=φL{ and M=Mλ(&M2 be decompositions of R-modules
ί = l
such that each Li is completely indecomposable, and let gji M-^Mj denote ike pro-
jection. Iff' L-+M is an isomorphism, then there exists a partition {1, •••, n} =
sucn tnat restriction map gjf: ζ&j.L^Mj is isomorphic for each j=l, 2.
324 H . ASASHIBA AND T . SUMIOKA
The following corollary is a generalization of Krull-Remak-Schmidt's
theorem.
n r
Corollary 1.5. Let M=®Li=®Mj be direct decompositions of an R-
module M with completely indecomposable modules L{. Then there exists a parti-
tion {1, •• , n } = / 1 Π ' Π Λ such that the induced map gji iV,—>My is ίsomorphic
and M=M
ι
® — @Mj(&Nj+ι(& — ®Nr for each ; = 1 , « ,r, where Nj=®IjLi
andgji M->Mj is a projection.
Proof. Regard ®M;- as Mk®{ 0 M ;), k=l, •••, r— 1. Then we get the
j=k j=k+l
assertion applying Lemma 1.4 inductively from case &=1 and/ is the identity
map.
Lemma 1.6. Let M=Q)ILi be a simple decomposition of a semi-simple R-
module M. Then for any direct decomposition M=M
ι
@M2 where M1 has a
finite composition length, there exists a partition / = / 2 U /2 such that M = ( φ / Lt)
Proof. Since M 1 C 0 / L t , we have MjC©//!,,- for some finite subset /'
of /. Put M'=®j,Li and Γ'=I—Γ. Since MXCLM\ M'=MX®M'2 (M'2=
M'ΠM2) and M2=M'2®M'2' for some submodules M'2 and MV of M2. It
is clear Λί=M / 0Mί / =Λf / 0(0 / //L | . ) . Applying Lemma 1.1 to Mf=®IfLi==
f
2y there exists a partition Γ=I1JIΓ2 such that M ' = 0 7 i L , . 0 M i =
Lt). Thus, for 72=/^ U/'' it holds 7=/ 1Π^2 and M = ( e / l i , ) θ M 2
Theorem 1.7. L^ί M=(BILi=(BJMj be completely indecomposable de-
compositions of an R-module M. Then for any finite subset J'={j\, '",j
n
} °f J>
there exists a subset Ir={i1, * ,4} °f 1 such that Lik^MJk for each k=l> ~ >n
and
/'My) = ( θ 7 ,
Proof. We see easily that the proof of Lemma 1.6 is valid in any com-
pletely reducible and Grothendieck category. Hence by the method of Harada
and Sai [3, Corollary 1, p. 334], the assertion holds.
n
EXAMPLE. Let M=ξBLi=ζ&JMj be completely indecomposable decom-
ί = l
positions of M andJ=J1 U / 2 a partition of/. Then by Krull-Remak-Schmidt's
theorem, for some subset I
λ
 of /={1, •••, n}y Λf=(0 / iZ/ l )0(0/2Λfy ). But it
is not necessarily satisfied that for I2=I—Ily M = ( 0 / i Λ ί y - ) 0 ( 0 / 2 L f ).
Let R be a field and M a vector space with dimension 3 over R. That is
M (=R?)={(αly α2y Λ3)'|Λ, ei?}, where (αly α2y α3)* expresses the transposed
matrix of (αly α2y α3). Put
KRULL-SCHMIDT'S THEOREM 325
^ = (1,1,1)', v2 = (1,1,0)', * 3 = (1,0,1)',
«! = (1,0,0)', «, = (0,1,0)', «, = (0,0,1)';
and L, =*>,•#, M1 = uiRy ι = l, 2, 3. Then M=L1φL2φL3=(M1®M2)φM3
and moreover
M = L
λ
®M2®M3 =
On the other hand, (L
2. Indecomposability of amalgamated sums
In this section, we assume a ring R is always right artinian and /^-modules
mean finitely generated right i?-modules except for Proposition 2.2.
Let (E) 0->K->L->M->0 be an exact sequence of i?-modules. We con-
sider the following condition:
(*) If X is a (non-zero) JR-module and <p: L-+X is a retraction (i.e. split
epimorphism) then there is no homomorphism ψ: M-+X such that <p=ψβ.
If an exact sequence (E) satisfies the condition (*), we say that (E) is a
(*)-sequence. Consider the following commutative diagrams of /?-modules
such that σ is a retraction and τ' is a section (i.e. split monomorphism):
(D)
X —
\ A
Z
(D') x-ϊ-
Λ
z
-* Y
A
In (D), let σ' be a right inverse of σ and put τ'=pσr. Then τf is a right in-
verse of T, and so we get the diagram (D'). Conversely, we can get (D) from
(D') and hence the condition (*) is equivalent to the following condition.
(*)' If X is a (non-zero) Λ-module and ψ': X-+M is a section, then there
is no homomorphism φ' \ X-+L such that ψ'=
REMARK 1. It is easy to see that (E) is a (*)-sequence if M has no direct
summand which is isomorphic to a direct summand of L. In particular, if
(E) does not split and M is indecomposable, then (E) is a (*)-sequence.
2. For an exact sequence (E), we can consider the following dual condi-
tion (**) of (*), and show the duals of all the results in Section 2 except for
Propositions 2.2 and 2.3.
(**) If X
x
 is a (non-zero) i?-module and φ
γ
: X
ι
-^L is a section, then
there is no homomorphism ψ^. XX-^K with φi=aψι.
n
Let L=(BLi be a direct decomposition of L into indecomposable modules
326 H . ASASHIBA AND T . SuMIOKA
Lh \<ί<n. Put S=EndR(L) and denote by J(S) the Jacobson radical of
S. Then every element φ in S is expressed by a matrix with coefficients
ψiji Lj^Li\ φ=(φij). As is well known, φ^J(S) if and only if each φi} is not
n
an isomorphism 1 <i, j <n. Moreover, let a=(aly ~yan) : ϋΓ-> 0 L f be a matrix
ι = l
expression of a: K-+L. Then the following proposition is immediate.
a β
Proposition 2.1. Let (E) 0->K-*L->M-+0 be an exact sequence and
n
L=(BLi a decomposition with indecomposable modules Lh \<i<n. Then the
* = 1
following conditions are equivalent.
(a) (E) is a (*)-sequence.
(b) {φ^S\φa=O}CZj(S).
(c) For φ = (φij) in S, each φu is not isomorphic, whenever <
REMARK 3. Monomorphisms a with the property (b) in Proposition 2.1
were investigated by Dickson and Kelly [2].
Let e and / be primitive idempotents of R and u{ an element of fje such
that u^fj2e for i = l , •••, n, where/ is the Jacobson radical of R. If an element
m of a module M and N is a submodule of M, a notation "rn^MjN" means
fn=m+N (GM/iV). Then for U^Je/βe, Ru^RflJf (simple). Moreover,
put Li=fRjfJ2y i=ly-- >n and K—eR/eJ. Consider (external) direct sum
n n
L=φLi and define a map α: iί->L by α t e ) = Σ w, ^ , where ea^eR/eJ=K
,=1 i=l
and uiea^fR/fJ2==Li. Now put M=Coker α. Then the following proposi-
tion is immediate from the definition of (*)-sequences. (See [5, Propositions
3.3 and 3.5] for Propositions 2.2 and 2.3.)
Proposition 2.2. Under the above notation, the following conditions are
equivalent.
(a) For ΰi(ΞjelJ2e, \<ί<n,we have Ru^--®Ra
n
(^Jejj2e.
(b) For ΰi^fjelfj2e9 l<i<n, ®i>m",un a™ linearly independent over a
division ring fRfjfJf (considering fjejfj2e as a left mldule).
(c) The exact sequence 0-+K->L->M->0 is a (*)-sequence.
Next, let Ll9 " ,Lr be indecomposable Λ-modules such that every homo-
morphism φ: Li-^Lj vanishes the socle of L{ for each pair i and j (iΦj), and
let K be a simple i?-module. Put Ni=Li
(ki) (&f-times direct sum of copies
of L^ and L=0JV f . Let a\\ K-^N^lKiKr), and a: K->L be maps with
ί = l
a=(a[, •••, a'r)\ and put M, = Cokera£ and M=Coker α. Then the following
proposition is immediate.
KRULL-SCHMIDT'S THEOREM 327
Proposition 2.3. Under the above notation, the following conditions are
equivalent.
{a) The exact sequence 0-+K->Ni-+Mi->0 is a (*)-sequence for each i =
l , - , r .
(b) The exact sequence 0^K->L-+M->0 is a (*)-sequence.
Let M be an i?-module. Recall that M is local (resp. colocaΐ) if M has
a unique maximal (resp. minimal) submodule. We denote the composition
length of Mby \M\.
n
Lemma 2.4. Let L=φLt and M=Mι(&M2 be decompositions of R-
ί = l
modules L and M> where Lfs are local {resp. colocaΐ), and πji M->Mj a projec-
tion for j=l, 2. If β: L-+M is an epimorphism (resp. monomorphism), then there
exists a partition {1, •••,#}= IχΠ_I2 such that each πβ\ L-^Mj induces an epi-
morphism (resp. monomorphism) nβ\ 0
τ
. L ^ M ) for j=l, 2.
Proof. We shall only show the assertion in case Lt 's are local, because
we can similarly do it in the other case. Let M denote the top MjMJ of M
and put N=σ(N) for every submodule N of M, where σ is the canonical homo-
mo rphism M->M. Then for some subset /' of {1, •••,#}, we have M—
0///8(Ll )=ilf10Af2. Using Lemma 1.1, as easily seen, there exists a partition
Γ=I[J±I'2 such that M=Σ/l//8(L |.)+Λf2=Σ/2//8(L ί)+M1. Then the assertion
is immediate from π1(M2)=π2(M1)=0.
Theorem 2.5. Let (E) 0->K->L -> M-^-0 be an exact sequence ofR-modules
n
such that L=(BLh L{ is local but is not simple and K is simple. Then the fol-
t = l
lowing conditions are equivalent.
(a) M is indecomposable.
(b) M has no direct summand which is isomorphίc to L t for some i=l, •••, n.
(c) The exact sequence (E) is a (*)-sequence.
Proof. We shall only prove that (c) implies (α), for the others are clear
(see Remark 1). Assume M is decomposable, say M=M1(BM2. By Lemma
2.4, there exists a partition {1, •••, w}=/1ΠΛ s u c n t n a t t n e restriction map
ψj: (BίfLi-^Mj of πjβ: φ L ^ M j is an epimorphism, 7=1, 2. But | L | = |ΛΓ|
ί = l
+ IK I — IMI + 1 . Hence ψj is an isomorphism for some j(j= 1 or 2), say j= 1.
n
Put φ=ψT1^iβ and let K: 0 / i L l - > 0 L f denote the canonical monomorphism.
Then φic is clearly the identity map of 0 ^ 1 ^ and so (E) is not a (*)-sequence.
REMARK 4. Theorem 2.5 is essentially due to Tachikawa [5, Lemma 1.1]
under Lemma 1.1. By Theorem 2.5 and Propositions 2.2 and 2.3, we can give
328 H . ASASHIBA AND T . SUMIOKA
simple proofs of Propositions 2.4, 3.1, 3.3, 3.5 in Tachikawa [4].
Proposition 2.6. Let L==φL, be an indecomposable decomposition and
a β
 ί = 1
(E) 0-~>K-+L^>M->0 be a (*)-sequence such that the n-th coordinate map
a
n
: K-+L
n
 of a is monomorphic. If Lly •••, Ln are colocal and Coker an is simple,
then M is indecomposable.
Proof. Put Lr=®Li and K
n
=a
n
(K). Consider the following diagram
with injection K ' and projection π
n
.
Then, since a
n
 (i.e. n
n
a) is monomorphic, the restriction map βr: L'-+M (i.e.
βκ
f) of β: L-+M is also monomorphic, and we have an exact sequence 0->
Lf->M->N-+0, where iV==Coker β\ It is easy to see that β(L')Γiβ(L
n
)=
β(K
n
) and so N=(β(L')+β{L
n
))lβ(L')^β(L
n
)lβ(K
n
). Therefore N is simple
(or zero) since L
n
/K
n
 is simple by the assumption. Suppose M is decomposable,
say M=M1φM2, and let 7Γ; : M-^-Mj denote the projection, 7 = 1 , 2. Then
by Lemma 2.4, there exists a partition {1, •••, n— 1 } = / I Π / 2 such that the re-
striction map ψk: ®IkLi->Mk of πkβ'\ L'-^M is monomorphic, Λ=l, 2. Since
\MI = \L'\ + \N\ < \L'\ + 1 , we may assume the monomorphism ψ
x
: ®IlLi
->M1 is isomorphic. Then by the same way as in the proof of Theorem 2.5,
we see that (E) is not a (*)-sequence. This verifies the assertion.
H
Proposition 2.7. Let L=®L{ and (E) be as in the above proposition. If
Ll9 ~ ,Ln are local and colocal and \Ln\divides |L, | for each i, (l<i<n), then
M is indecomposable.
Proof. Suppose M is decomposable, say M=M1®M2. Let πk: M->Mk
denote the projection, k=ίf 2. As in the proof of Theorem 2.5, for some
partition {1, •••, n}=J1]J[J2y the restriction map πkβ: 0/jfeLx-^MΛ of πkβ: L->Mk
is epimorphic for each k=l9 2. On the other hand, as in the proof of Proposi-
tion 2.6, for some partition {I, "-yn—l}=I1JJI2, the restriction map πkβ:
®IkLi->Mk of πkβ: ®Li-*Mk is monomorphic. Put d = \Li\. Then the
ί = l
above fact implies Σ / ^ , ^  \Mk\ <Σljkci, a n d hence (Σ/^»)/^n^(Σjk^i)lcn for
k=ly 2. But cn divides ct for each ί = l , ••-, n and (Σ/ 1^ί)/ ί :ι.+(Σ!/ 2^)/^+l =
( Σ ^ O K ^ ί Σ / ^ O K + ί Σ / ^ O K This shows (ΣlikCi)ICn=(ΣjkCi)lc. for s o m e
ί — 1
k=l or 2, say k=l. Hence Σ / ^ ί — l ^ i l — Σ / ^ ί a n ( l the monomorphism
KRULL-SCHMIDT'S THEOREM 329
7tφ: ®IiLi->M1 is isomorphic. Thus (E) is not a (*)-sequence. This verifies
the assertion.
References
[1] G. Azumaya: Corrections and supplementaries to my paper concerning Krull-Remak-
Schmidt's theorem, Nagoya Math. J. 1 (1950), 117-124.
[2] S.E. Dickson and G.M. Kelly: Interlacing methods and large indecomposables,
Bull. Austral. Math. Soc. 3 (1970), 337-348.
[3] M. Harada and Y. Sai: On categories of indecomposable modules I, Osaka J. Math.
7 (1970), 323-344.
[4] H. Tachikawa: On rings for which every indecomposable right module has a unique
maximal submodule, Math. Z. 71 (1959), 200-222.
[5] H. Tachikawa: On algebras of which every indecomposable representation has an
irreducible one as the top or the bottom Loewy constituent, Math. Z. 75 (1961), 215-
227.
Department of Mathematics
Osaka City University
Osaka 558, Japan



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On Krull-Schmidt's theorem and the indecomposability of amalgamated sum

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