SUMIOKA, TAKESHI

Osaka City University Repository

Sumioka, T.
Osaka J. Math.
12 (1975), 449-456
A CHARACTERIZATION OF THE TRIANGULAR
MATRIX RINGS OVER QF RINGS
TAKESHI SUMIOKA
(Received May 8, 1974)
In [4], Harada proved that a ring R is a right QF-3 and semi-primary
hereditary ring if and only if R is a direct sum of rings whose basic rings are
the rings of triangular matrices over division rings. We consider an analogous
result to the above one for a right QF-3 semi-primary ring with some injective
properties. By Zaks [8], the ring R of triangular matrices of degree n^2 over
a QF ring has the injective dimension one both as right and left 7?-modules,
and moreover it is easy to see that the ring R is a QF-3 ring whose maximal
right quotient ring is a QF ring. It is our purpose to show that for a basic
indecomposable semi-primary ring, the converse is also true.
The author wishes to express his appreciation to Professor M. Harada who
has carefully checked my original manuscript and has given the author valuable
advices.
Throughout this paper we shall assume that every ring R has an identity
element 1, and every Λ-module is unitary. The notations MR and #Mare used
to underline the fact that M is a right or a left 7?-module, respectively. For a
ring R, a right (resp. left) Λ-module M is called a minimal faithful module if
M is a faithful J?-module and every faithful right (resp. left) jf?-module contains
an isomorphic image of M as a direct summand. A ring R is called a right
(resp. left) CλF-3 ring if R has a minimal faithful right (resp. left) module, and
R is called a QF-3 ring if R is both a right and left QF-3 ring.
For a semi-primary ring R, the following conditions are equivalent, (see
Jans [5]).
(1) R is a right QF-3 ring.
(2) R has a faithful projective injective right ideal.
Let S be a ring which contains a ring R as a subring. Then S is called
a right (resp. left) quotient ring of R if S is a rational extension of R as a right
(resp. left) jR-module. By R' we denote the maximal right quotient ring of R.
If J? is a QF-3 ring, then the maximal right quotient ring of R coincides with
the maximal left quotient ring of R, (see Tachikawa [7]).
Lemma 1. Let S be a QF ring. Let R be a right QF-3 ring such that R
450 T. SUMIOKA
is a subring of S and an essential submodule of S as a right R-module, and let eR
be a minimal faithful right ideal of R, where e is an idempotent in R. Then we
have S=Hom
eRe(eR, eR), and S is infective both as right and left R-modules.
Therefore, S is the maximal left quotient ring of R.
Proof. Since eR is jR-injective and eR is an essential submodule of an
72-module eS, we have eR=eS, and hence eS is right S-faithful because of
assumptions that eR is right JR-faithful and RR is an essential submodule of SR.
Therefore, eRe=eSe is a QF ring and eR=eS is a finitely generated projective
left eRe-moάule, (see Curtis and Reiner [3]), which shows that Hom
eRe(eR, eR)
is right 72-injective since eR is right Λ-injective. Moreover we have
S^Hom
eSe(eS,eS)=HomeRe(eR,eR)y because S is QF. Therefore S is right
7?-injective. Moreover, since eR is left ^ίte-injective, S is also left Λ-injective,
(see Cartan and Eilenberg [2], Chap. VI, Prop. 1.4). On the other hand,
Hom
eRe(eR, eR) is a left quotient ring of R, and consequently S is the maximal
left quotient ring of R. This completes the proof.
Let R be a right QF-3 ring whose maximal right quotient ring R' is QF.
Then, by Lemma 1,1?' is a injective hull of R both as right and left 72-modules
and R' is the maximal left quotient ring of R.
For a ring 7? we consider the following conditions:
(Cj) R is a right QF-3 ring whose maximal right quotient ring R' is QF,
and the injective dimension of R as a right Λ-module is one.
(C2) R satisfies the condition Cn and moreover the injective dimension
of R as a left Λ-module is one.
We study a semi-primary ring R satisfying the condition C2. Now let R
be a semi-primary ring, and let l=Σ^*y ^e a decomposition of 1 into a sum of
orthogonal primitive idempotents ei} such that e{jR^eklR if and only if i=k.
If we write e=^e
ίl9 then eRe is a basic ring.
The following lemma is obtained from Morita [6], the proof of Theorem
1.1.
Lemma 2. (Morita) Let R be a right QF-3 semi-primary ring with a
minimal faithful right ideal eR, where e is an idempotent in R. If eRe is a left
artinian ring and eR is a finitely generated left eRe-module, then R is left QF-3.
By Lemma 1 and Lemma 2, we have
Corollary 1. Let R be a right QF-3 semi-primary ring with minimal faith-
ful right ideal eR. Then the following conditions are equivalent.
(1) There exists a QF ring S such that R is a subring of S and RR is an
essential submodule of SR.
(2) The maximal right quotient ring R' is QF.
TRIANGULAR MATRIX RINGS 451
(3) The double centralizer Hom
eRe(eR, eR) of eRR is QF.
Moreover if R satisfies these equivalent conditions, we have S=R'=
Rom
eRe(eR, eR) and R is (left) QF-3.
Proposition 1. Let R be a semi-primary ring, and let eRe be a basic ring
of R where e is an idempotent in R. Then R satisfies the condition C1 (resp. C2)
if and only if eRe satisfies the condition C
x
 (resp. C2).
Proof. Suppose that R is a right QF-3 ring with a minimal faithful right
ideal e'R of R. Then we may assume that e' is an idempotent in eRe since eR
is Λ-faithful. On the other hand, eRe is Morita equivalent to R with respect
to eR. Therefore, e'R®RRe^e'Re is a minimal faithful right ideal of eRey and
so eRe is right QF-3. If R' is QF, then e'Re' is QF and e'Re is a finitely gen-
erated faithful projective left e'Re' -module and so }lom
e
,Re,(e'Re, e'Re) is QF.
Thus, by Corollary 1, the maximal right quotient ring of eRe is QF. Using
the fact that
 RRe is a generator, it is similarly proved that if eRe is a right QF-3
ring whose maximal right quotient ring is QF, then so is R. The remaining
assertions are easily showed, because eRe is Morita equivalent to R with respect
to eR.
From Proposition 1, we may assume our ring is basic.
Lemma 3. Let R be a semi-primary basic ring satisfying the condition CΊ
and let e be an idempotent such that eR is a minimal faithful right ideal in R, and
set f=l—e. Then fRe=0.
Proof. As in the proof of Lemma 1, eRe is a QF ring and eR=eR' is left
^Rtf-projective. And moreover eR' is a minimal faithful right ideal in R'.
Consider an exact sequence
Then this sequence induces an exact sequence
0 -> HomR(eR, R) -> HomR(eR, R') -> HomR(eR, R'/R) -^  0 .
Since R'/R is a finitely generated injective right J?-module and eR is a projective
left tf/fe-module, HomR(eR, R'/R) is finitely generated right eRe-injective, which
implies that Hom#(&R, R'/R) is right ^Jfe-projective, (see Curtis and Reiner [3]).
Therefore the exact sequence above splits, and consequently we have a
right ^^-isomorphism Re®M^R'e with some right eRe-module M, since
Hom^&R, R)^Re and HomR(eR, R')^R'e. This isomorphism induces an iso-
morphism Re®
eReeR®MeReeR^R'e®eReeR as right JR-modules. On the other
hand, we have R'e^}ίlom
eRe(eRy eRe) and moreover HomeRe(eR, eRe)®eReeR~
Hom
eRe(eR,eR)^R', by Auslander and Goldman [1]. Therefore R'e®eReeR^R'
452 T. SUMIOKA
and so that Re®
eReeR^ReR. Thus, the fact that R' is right JR-injective implies
that ReR is also right 7?-injective.
Now suppose fRe^pO. Then there exists a primitive idempotent /,• such
that fiRe^Q and f{R is a direct summand/R. Since f{ReR is right .R-injective
and/
z
 7? is an indecomposable right jR-module, we have fiReR=fiR. Therefore
ffR is right .R-injective and/
z
j? is isomorphic to a direct summand of eR'=eR.
This is a contradiction, because R is a basic ring. Thus we have/Re—0.
Lemma 4. Let R be a ring and let M and N be right R-modules such that
N is a submodule of M. Let f be an idempotent in R such that fR is left fRf-pro-
jective. If M/N is right R-injective then Mf/Nf is also right fRf-ίnjective.
Proof. Since HomR(fRy M/N)^Mf/Nf, Lemma 4 is immediate.
Corollary 2. Let R, e and f be as in Lemma 3. Then fR'f is a right fRf-
ίnjective hull offRf, and fRf has the injectίve dimension ^ 1 as a right fRf-module.
Moreover fR'f is a QF ring.
Proof. Since fRe=Q, fR is left/R/-projective. It is obvious that fR' and
fR'IfR are right Λ-injective. Therefore, by Lemma 4, fR'f and fR'f/fRf are
right /R/-injective. Moreover, fRf is an essential submodule of a right fRf-
module fR'f, because fRe=0 and fR is an essential submodule of a right R-
module fR'. Thus, fR'f is a right /R/-injective hull of fRf. Next, setting
S=eR'+fR'f, clearly S is a ring. Since R is right QF-3 and RdSaR', S is
also right QF-3 and R'=Sf. Hence, by Lemma 1, R' is right S-injective, and
consequently fR'f is a QF ring, by Lemma 4.
Lemma 5. Let S be a QF ring such that fSe=Q, where e and f are idem-
potents in S and e+f=l. Then, we have eSf=Q.
Proof. For a right ^-module MR, we denote the socle of M#by Soc(M/?).
Suppose that eSf^O. Then, there exist idempotents ef and f. such that
Soc(eiSs)fJ φ 0, and ei+e'=e and f.-\-f=f are sums of orthogonal idem-
potents, because eSf is a right ideal of 5. By our assumption, S is a
QF ring, which implies that Soc(e
z
 5) is a simple right S-module. Hence
Soc(e£S)s^(fJ S/fJN)Sy where ./V is the Jacobson radical of S.
On the other hand, by Lemma 4, fSf is also a QF ring, hence there exists
an idempotent/£ in fSf such th.ztfjSf/fjNf^Soc(fkSffSf) as right /^/-modules.
But we hzvefSe=Q and this shows that/y5//yΛ/r^Soc(/^iSίs) as right S-modules,
and consequently Soc(e
z
 ASr
s
)^Soc(jfA:S's). Thus we have e^^f^s, since these
are injective hulls of Soc(et ίS's) and Soc(fkSs), respectively. This contradicts
the assumption fSe=Q.
Lemma 6. Let R, e and f be as in Lemma 3 and moreover let R be an
TRIANGULAR MATRIX RINGS 453
indecomposable ring. Then Rf is a faithful left R-module, therefore fR' is a
faithful right R'-module.
Proof. By Corollary 1, R is a left QF-3 ring. Let Rg be a minimal faith-
ful left ideal of R where g is an idempotent. Then Rg=R'g and R'g is a mini-
mal faithful left ideal of R', hence gR'^eR. Since fRe=0 and/φO,
 RRg is
not isomorphic to any submodule of
 RRe. If RRg is isomorphic to a submodule
of
 RRf, then Rfis clearly a faithful left jR-module.
Now suppose that
 RRg is not isomorphic to any submodule of RRf. Then,
we may assume thatg=e"-\-f and e=e"+e' are sums of (non-zero) idempotents
e'', /' and e", e', respectively, where e1', e"^eRe and/'e//?/. Since eR' is a direct
sum of non-isomorphic indecomposable right ideals of R't we have f'R'^e'R'.
On the other hand, fR'e"=fRe"=Q and in particular f'R'e"=Q, because R'e" is
a direct summand of a left Λ'-module Kg. Hence, we have e'R'e"—Q and
consequently (l—e")R'e"=Q. Therefore, by Lemma 5, we have e"R'(\— e")=Q.
But e" is in R. It follows that R is decomposable as a ring, which is a con-
tradiction. Thus, Rg is isomorphic to a submodule of /?/ and Rf is a faithful
left 7?-module. Then, obviously,//?' is a faithful right 7?'-module.
Lemma 7. Let R be a ring, and let e and f be idempotents in R such that
fRe=0 and e-\-f=l. Let M and N be left R-modules such that N is a submodule
of M and eM=eN. If M/N is left R-ίnjectίve, then /M///V is also left fRf-in-
jectίve.
Proof. For each x<=R and fm+fN<= fM/fN, we set x(fm+fN)=fxfm+fN.
Then, since fRe=Q, fM/fN is regarded as a left Λ-module, which, restricting
its operation to fRf, coincides with the original left /^/-module fM/fN. Next,
define a map M/N-* fM/fN by corresponding each m + N^M/N to
fm-\-fNξ=: fM/fN. Then, this map is obviously a left Λ-isomorphism, because
of our assumptions fRe—0 and eM=eN. Therefore, fM/fN is left Λ-injective.
Now we show that fM/fN is left /R/-injective. Let / be an arbitrary left
ideal of fRf, and let φ be an arbitrary /R/-homomorphism of / into fM/fN.
Then, using the map φ, we define a map Λ/Γ of a left Λ-ideal RI into a left
Λ-module fM/fN as follows: for each x^RI, set (x)ψ>=(fx)φ. Then it is clear
that -v/r is an J?-homomorρhism. Since fM/fN is left Λ-injective, there exists an
element fm+fN in fM/fN such that for each x<=RI, (fx)φ=x(fm+fN). In
particular, for any element a in /, (a)φ=a(fm+fN), which implies fM/fN is left
/R/-injective.
Lemma 8. Let R be a semi-primary basic indecomposable ring satisfying
the condition C2 and let e be an idempotent such that eR is a minimal faithful right
ideal of R, and set f=l—e. Then fRf is a (right and left) ζλF-3 ring.
454 T. SUMIOKA
Proof. Let Rg be a minimal faithful left ideal of R where g is an idem-
potent. Then, by Lemma 6, we may assume that g is in fRf. Thus, fRg is a
faithful projective left ideal of fRf. On the other hand, we have Rg=RR'g^RR'e,
hence fRg^fR'e as left /7?/-modules. However, by the assumption C2 and
Lemma 7, fR'ejfRe is left /R/-injective. Therefore, fR'e is left /R/-injective,
since fRe—Q. Consequently fRf is a left QF-3 semi-primary ring with a mini-
mal faithful left ideal fRg, because of fRg^ fRgj for /φ/. Moreover, gRg is
a right artinian ring and fRg is a finitely generated right gRg-module. Thus,
by Lemma 2, fRf is a right QF-3 ring.
Corollary 3. Let Ry e andf be as in Lemma 8. Then fRf has the injectίve
dimensional as a left fRf-module.
Proof. By Corollary 2, Lemma 8 and Lemma 1, fR'f is left //?/-injective.
On the other hand, by Lemma 7, fR'f I fRf is left /R/-injective. Therefore, fRf
has the injective dimensional as a left/R/'-module.
Corollary 4. Let R, e and f be as in Lemma 8. Then fRf is an indecom-
posable ring.
Proof. Setting S=fRf and T=fR'f, then S is a right QF-3 ring and T is a
QF ring by Lemma 8 and Corollary 2 (or Lemma 6), respectively. If f'S is
a minimal faithful right ideal of S, f'S=f'T is a minimal faithful right ideal of
T, where /' is an idempotent. However, T is Morita equivalent to R' with
respect to/7?7 and sof'R' is a minimal faithful right ideal of R'. Now suppose that
S is a decomposable ring, i.e. there exist orthogonal central idempotents g and h
in 5 such that f=g-\-h. Then we may assume that f'=g'-\-h' for some (non-zero)
idempotents g'^gRg and h'(=hRh. Because of f'S =f'T= fR'f, we have
g'R'h=g'Sh=Q and h'R'g=h'Sg = Q. Since f'R'^eR', there exist orthogonal
idempotents #' and e" in eRe such that e'R' ~g'Rr, e"Rr ~h'R' and e=£'-|-e".
Then, noting fRe = 0, we have (e'+£)#(e" + A)=0 and (e"+h)R(e'+g) = Q.
This contradicts the indecomposability of R. Thus S is an indecomposable ring.
Theorem 1. Let R be an indecomposable basic ring. If R is a right QF-3
semi-primary ring whose maximal right quotient ring is QF and R has the injective
dimension 1 both as right and left R-modules, then R is ίsomorphίc to the ring of
triangular matrices of degree n^2 over a QF ring. Therefore, R is (right and left)
artinian. The converse is also true.
Proof. Let e be an idempotent such that eR is a minimal faithful right
ideal in R. Set e1=^e, /i = 1 and f2=f1—e1= 1—eλ. Now assume that there
exist idempotents el9 •••, £/_! and fly ••-,/,-, which satisfy the following
conditions:
TRIANGULAR MATRIX RINGS 455
{ej} is mutually orthogonal.
f.= \—(el-\ h^ -ι)> in particular /i=l.
I' is right 7?-faithful.
is an indecomposable basic semi-primary ring and an essential
submodule of fiR'fi as a right/tJ?/r module.
(2) fiRfi satisfies the condition C2.
(3) *,_!/?/,-!=*,-!*'/,-!, e^R'^eR' and/^.^O.
Let e{ be an idempotent such that βfRff is a minimal faithful right ideal in fiRfi
and set/,•+!=/,—£,•. Then, from (1) and (2), we can easily show that the idem-
potents e19 •••, £2 and /ι, ,/t +ι satisfy the conditions (1) and (3) on which we
replace i by ί+1. And moreover f
ί+1Rfί+1 is either a ring satisfying the con-
dition C2 or a QF ring.
Thus, for some n^2, there exist idempotents ely •• ,e l land/1, ,/„ such that
f
n
Rfn is a QF ring and these idempotents satisfy the conditions (1) and (3) for
each i, 2^i^n-\-l, where /M+1—0. On the other hand eRe is a QF ring. It
follows that R is isomorphic to the ring of triangular matrices of degree n over
a QF ring eRe. Therefore, R is a (right and left) artinian.
Conversely, we assume that R is isomorphic to the ring of triangular ma-
trices of degree n^2 over a QF ring. Then, by Zaks [8], R has the injective
dimension 1 both as right and left jR-modules. Moreover, it is obvious that
Jf? is a (right) QF-3 ring whose maximal right quotient ring of R is QF.
Now let R be a semi-primary ring, and let R=R^ - ®R
n
 a decomposition
of R into indecomposable rings Rj with basic rings eiRe{. If R is a right QF-
3 ring whose maximal right quotient ring is QF and R has the injective dimen-
sion ίg 1 both as right and left Λ-modules, then so does R^ or equivalently efte^
for each i (see Proposition 1). And moreover the converse is also true. There-
fore, by Theorem 1, we have
Corollary 5. Let R be a ring. Then R is a right QF-3 semi-primary ring
whose maximal right quotient ring is QF and R has the injective dimension ^  1
both as right and left R-modules if and only if R is a direct sum of rings whose basic
rings are isomorphic to the rings of triangular matrices over QF rings.
OSAKA CITY UNIVERSITY
456 T. SUMIOKA
References
[1] M. Auslander and O. Goldman: Maximal orders, Trans. Amer. Math. Soc. 97
(1960), 1-24.
[2] H. Cartan and S. Eilenberg: Homological Algebra, Princeton Univ. Press, 1956.
[3] C.W. Curtis and I. Reiner: Representation Theory of Finite Groups and Associa-
tive Algebras, Interscience, New York, 1962.
[4] M. Harada: QF-3 and semi-primary PP rings I, Osaka J. Math. 2 (1965), 357-368.
[5] J.P. Jans: Protective injective modules, Pacific J. Math. 9 (1959), 1103-1108.
[6] K. Morita: Duality in QF-3 rings, Math. Z. 108 (1969), 237-252.
[7] H. Tachikawa: Quasi-Frobenius Rings and Generalizations, Lecture Note in
Math. 351, Springer, Berlin, 1973.
[8] A. Zaks: Injective dimension of semi-primary rings, J. Algebra 13 (1969), 73-86.


A characterization of the triangular matrix rings over QF  rings

http://dlisv03.media.osaka-cu.ac.jp/infolib/user_contents/sugaku/111F0000002-01202-17.pdf

A characterization of the triangular matrix rings over QF rings

Abstract

Similar works

Full text

Available Versions

Osaka City University Repository