KURATA, YOSHIKI

MORIMOTO, SHOJI

Kurata, Yoshiki

Morimoto, Shoji

Institutional Repositories DataBase (IRDB)

A note on quasicorational extensions

Osaka City University Repository

Kurata, Y. and Morimoto, S.
Osaka J. Math.
15 (1978), 683-692
A NOTE ON QUASI-CORAΊΊONAL EXTENSIONS
YOSHIKI KURATA AND SHOJI MORIMOTO
(Received September 13, 1977)
In his paper [3] Bland has introduced quasi-corational completions of
modules and by means of this concept he has given a characterization of finite
direct sums of complete matrix rings over perfect completely primary rings.
In this note we shall look at his results again from the point of view of
torsion theories. By this we can see his results in perspective and give simple
proofs of some of his results. In section 1 we shall define an idempotent pre-
radical tQ of mod-i? for a fixed i?-module Q. As is well-known, when Q is
projective, tQ is a radical. An example will be given to show that tQ is not
always a radical even if O is quasi-projective. However we can show that, for
submodules of a quasi-projective module Q, tQ acts as a radical (Proposition 1.2).
Using this proposition, in section 2, we shall give a slightly simple char-
acterization of a quasi-corational extension of an i?-module (Theorem 2.4). We
shall show that a maximal quasi-corational extension of an i?-module can be con-
structed naturally by this theorem and the proposition.
In section 3 we shall treat quasi-corationally complete modules. In section
4 we shall only deal with i?-modules M having projective covers 0-»JK->P-*M
—•0. Wu and Jans [9] has shown how to construct a quasi-projective cover
of M out of the projective cover of M. They showed that P modulo the unique
maximal i?-End#(P)-submodule contained in K is the desired quasi-projective
module. We shall show that this submodule is precisely P(0: M) (Proposition
4.1). We also show that the quasi-corationally completeness of M can be de-
scribed by means of the radical tP (Theorem 4.6). Finally in closing this note
we shall give an example of ar ϋ-module which is quasi-corationally complete
but not quasi-projective.
Throughout this note i? will denote an associative ring with identity and
all modules will be unital right i?-modules. The category of unital right R-
modules is denoted by mod-i?. For the notions and terminologies about torsion
theories we refer to Stenstrϋm [8].
1. Preliminaries
A subfunctor t of the identity functor of mod-i? is called a preradical of
684 Y. KURATA AND S. MORIMOTO
mod-i?, i.e., t assigns to each module M a submodule t(M) of Min such a way that
every homomorphism M->N induces t(M)->t(N) by restriction. A preradical
t is idempotent if t(t(M))=t(M) for every module M and is called a radical if
ΐ(M/t(M)) = 0 for every module M.
To each preradical t we can associate two classes of modules, namely
T{t)={AR:t{A) = A} and F(t) = {BR: t{B) = 0} .
T(t) is closed under homomorphic images and direct sums and dually F(t) is
closed under submodules and direct products.
Now let Q be a module and define a preradical £# of mod-i? as
h(M) = Σ {Im(f):fεΞHomR(Q, M)}
for each module M. Then ί0 is idempotent and tQ(Q)=Q. Moreover tQ is a
unique minimal one of those preradicals t of mod-jR for which t{Q)—Q.
A module M belongs to T{tQ) if and only if M is a homomorphic image
of a direct sum of copies of Q. While M belongs to F(tQ) if and only if
HomR(Q, M)=0. F(tQ) is also closed under group extensions and becomes a
torsion-free class.
If in particular Q is a projective module, then tQ becomes a radical and
F(ΐQ) is closed under homomorphic images, as is easily seen. However this is
not true in general. The following example shows that tQ is not always a radical
even if Q is a quasi-projective module.
EXAMPLE 1.1. Let R be a right Artinian ring with identity and N (Φθ) its
Jacobson radical. There exists an integer n>0 such that Nn=0 and iV^φO.
We put Q=RINn-\ Then, since f(N)czN for all / eEnd(12*), by [9, Proposi-
tion 2.1] we see that Q is quasi-projective. Furthermore N"'1 is not a direct
summand of R and so Q can not be a projective module. It is easy to see that
tQ(R) = AnnR(Nn-1)y the left annihilator of N11'1 in R, and tQ(R)φR. Since
JVn=:0, we have Nn~1dNc:tQ(R.) and so, by considering the canonical homomor-
phism Q->RltQ(R), we see that tQ(RltQ(R))=RltQ(R)Φθ.
We note that, for a module (), £Q is a radical if and only if T(tQ) is closed
under group extensions (see e.g. [2, Proposition 3] or [7, Proposition 1.1]). So
the above example also shows that, in contrast with F(tQ), T(tQ) is not always a
torsion class in general.
As we have shown above, tQ is not always a radical, but for submodules of
Q it acts as a radical.
Proposition 1.2. Let Q be a quasi-projective module and K and K' sub-
modules of Q such that Kf c K . Then tQ(KIK')=0 if and only if tQ(K)dK'. In
particular we have tQ(KltQ(K))=0.
A NOTE ON QUASI-CORATIONAL EXTENSIONS 685
Proof. Suppose that tQ(KIK')=Q. Then we have (tQ(K)+K')lK'ϊΞF(tQ).
On the other hand, since tQ is idempotent, tQ(K)^ T(tQ) and so (tQ(K)+Kr)jKf
GΞ T(tQ). Thus we have (tQ(K)+K')IK'=0 and tQ(K)czK'. Conversely sup-
pose that ί ^ c Γ and claim that HomR(Q, KIK')=0. Take/eHom^(Q,
K/K'). Then there exists a homomorphism φ: Q-+Q making the diagram
QIK'
commutative, where Q->Q\Kf and K\Kf -+Q\Kf are the canonical map and the
inclusion map respectively. For any x^Q, φ(x)-\-K'=f(x)=x/-{-K/ for some
^ G l From this it follows that φ(x)<=K and hence φ(x)<=tQ(K). Therefore
f(x)=φ(x)-]-K/=0 by assumption. Thus we have/=0.
2. Quasi-corational extensions
We now record some required definitions.
A quasi-projective cover of a module M is an exact sequence Q-+M-+0
with the properties that Q is quasi-projective, Ker(/3) is small in Q and, for
every nonzero submodule Q' of Ker(/3), Q\Qf is not quasi-projective.
An exact sequence iV-^M—>0 of modules is called corational by M if every
factor module of Ker(/) belongs to F(tN). It is called quasi-corational by M
if it is corational by M, besides M has a quasi-projective cover Q-+M-+0 and
there exists a homomorphism φ: Q-+N such that fφ=β. Note that, since
Ker(/) is small in iV (see [4, Theorem 2.3]), φ must be an epimorphism.
First we shall quote [2, Proposition 2] in a slightly general form.
Lemma 2.1. Let 0-^L->iV->M->0 be an exact sequence of modules such
that L is small in N and let A be a module. If every factor module of A belongs
to F(tM)y then A belongs to F(tN). In the case where F(tN) is closed under factor
modules of A, the converse is also true.
Proof. We claim that Hom^iV, 4^) = 0. Take any ^
Theng induces the mapping g*: M->A/g(L) given by g*{f(x))=g(x)+g(L) for
x^N. Since £ * = 0 by assumption, g(N)=g(L) and hence iV— L+Ker (g).
However L is small in N and so N=Ker (g). Thus we have £ = 0 . The latter
half of the lemma follows from the fact that F(tN)aF(tM).
As a consequence of this lemma we have
686 Y . KURATA AND S. MORIMOTO
Corollary 2.2. Let 0->L-+N->M->0 be an exact sequence with L small
in N. If F(tM) is closed under factor modules, then F(tN)=F(tM) holds.
From Proposition 1.2 we have
Corollary 2.3. Let Q be a quasi-projective module and K a submodule of O.
Then every factor module of K belongs to F(tQ) if and only if K itself belongs to
F{tQ).
Proof. Note that if K^F{tQ)y then F(tQ) is closed under factor modules
of K by Proposition 1.2.
We now obtain a characterization of quasi-corational extensions.
Theorem 2.4. An exact sequence N->M-*0 is quasi-corational by M
if and only if
(1) Mhas a quasi-projective cover 0->ίΓ-><3->M-»0,
(2) there exists an epίmorphism φ: Q-^N such thatfφ=β, and
(3) K e r ( / ) e F ( ί
β
) .
Proof. The "only if" part follows from the definition and Lemma 2.1, so
we only show the "if" part. Assume that (1), (2) and (3) hold. Then Ker(φ)
is a submodule of K and the diagram
/
commutes, where /3* and φ* are homomorphisms induced by β and φ respec-
tively. Hence Ker(/) is isomorphic to R^/Ker(φ) and by Proposition 1.2
tQ(K)aKer(φ). Every factor module of Ker (/) is also isomorphic to KjK' for
some submodule Kr of K containing Ker(φ). Again by Proposition 1.2
and thus every factor module of Ker(/) belongs to F(tQ). Since
F(tQ)cF{tN)y this implies that N-+M-+0 is quasi-corational by M.
As is seen from the proof of the theorem, every exact sequence which is
quasi-corational by M is of the form
β*
for some submodule K' of K, where 0-^K-^Q—>M—>0 is a quasi-projective
cover. Conversely for a module M having a quasi-projective cover 0—>.K—>£)
β β*
->M->0, by Theorem 2.4 an exact sequence Q\K' >M-^0 with K' a sub-
A NOTE ON QUASI-CORATIONAL EXTENSIONS 687
module of K is quasi-coratinoal by M if and only if KIK'=Ker(β*)GF(tQ),
or equivalently, by Proposition 1.2, tQ(K)dK'.
Consequently, in particular,
Q/tQ(K) - - * M -* 0
is surely quasi-corational by M, where β denotes the homomorphism induced
β*
by β, and moreover, for each exact sequence Q\K' > M-+0 which is quasi-
corational by Λf, the mapping φ: QjtQ{K)-^QIKr defined by φ(x+tQ(K)) =
x-\-K' makes the diagram
4
commutative, i.e.,
QjtQ{K)—>M->0
is maximal quasi-corational by M in the following sense.
An exact sequence iV-»M->0 of modules is called maximal quasi-corational
by M if it is quasi-corational by M and, for each exact sequence N'-*M-*Ό
which is quasi-corational by M, there exists a homomorphism φ: N->N' such
Ul3tf'φ=f.
In this definition, if a homomorphism φ': N-+N' also satisfies that f'φ'=fy
fthen we have Im(φ—φ')cKer(/'). Since iV'—>M—>0is (quasi-) corational,
every factor module of Ker (/') belongs to F(tN') and whence to F(tM). By
Lemma 2.1 Ker (/') itself belongs to F(tN). Therefore φ—φ': N->N' is a zero
mapping and thus we have φ=φ'. From this it follows that if N->M->0 and
iV-»M—>0 are both maximal quasi-corational by M, then there exists an iso-
morphism φ: N-+N' for which the diagram
N — M
is commutative.
688 Y. KURATA AND S. MORIMOTO
Corollary 2.5 ([3, Theorem 2.1]). For a module M having a quaisi-pro-
jective cover 0 —> K -> Q -> M —• 0, £/?£ sequence
Q/tQ(K)—*M->0
is maximal quasi-corational by M.
3. Quasi-corationally complete modules
A module M is called quasi-corationally complete if N->M->0 is quasi-
corational by M, then/is an isomorphism.
In case M has a quasi-projective cover, the exact sequence M—>M-^0 is
certainly quasi-corational by My where 1 means the identity map of M. Hence
we have
Theorem 3.1. For a module M having a quasi-projective cover, the following
conditions are equivalent:
(1) M is quasi-corationally complete.
(2) M—>ikf —>0 is maximal quasi-corational by M.
(3) Hom^(M, —) is right exact on all exact sequences of the form 0—*L-*
iV—>M-»0 which is quasi-corational by M.
(4) Every exact sequence N-+M-+0 which is quasi-corational by M splits.
Proof. (1)=>(2). If N->M->0 is quasi-corational by M, then / must be
an isomorphism by assumption. Hence the diagram
M >M
<*\
1is commutative. This shows that M—>M—>0 is maximal quasi-corational by M.
(2)=#>(3). If iV->M-->0 is quasi-corational by M, then by the maximality
of Λf-*Λf->0 we can find a homomorphism φ: M->N such that the diagram
M > M
*[
N • M
A NOTE ON QUASI-CORATIONAL EXTENSIONS 689
is commutative. Therefore, for any g ^HomR(M, M), f(φg)=g and thus
Hom^(M, —) is right exact on 0->L-*N->M-+0.
(3)=§> (4). This is clear.
(4)=Φ(1). If iV->Λf—>0 is quasi-corational by M, then by assumption
Ker(/) is a direct summand of N. However Ker(/) is small, it must be zero,
i.e., M is quasi-corationally complete.
We also have another characterization of quasi-corationally complete
modules by means of tQ.
Theorem 3.2. For a module M having a quasi-projective cover 0-^K-^Q
—>M—>0, the following conditions are equivalent:
(1) M is quasi-corationally complete.
(2) ^
(3)
Proof. By Corollary 2.5, 0-+KltQ(K)->Q!tQ(K)-+M->0 is maximal
quasi-corational by M. In this case, K G T(tQ) if and only if ^ is an isomor-
phism, and moreover this is so if and only if M—>M—>0 is maximal quasi-
corational by M since the maximal quasi-corational extension is unique up to an
isomorphism. Thus (1), (2) and (3) are equivalent by Theorem 3.1.
It follows from this theorem that a quasi-projective module is quasi-cora-
tionally complete ([3, Theorem 1.1]), but the converse of this fact is not always
true in general, as we shall show later.
The following is a generalization of [3, Theorem 2.3].
Proposition 3.3. Let Q be a quasi-projective module and K its submodule
such that K is small in Q and K^ T(tQ). Then HomR(QIK, —) is right exact on
all exact sequences of the form 0->L-*QIK->N->0 which is corational by N.
Proof. Take any geHom
Λ
(Q/K, N). Since Q is quasi-projective, there
exists a homomorphism φ': Q-+Q for which
is commutative, where π denotes the canonical homomorphism Q-^QjK. We
put φ"—πφ'. Then to prove the proposition it is sufficient to show that
690 Y. KϋRATA AND S. MORIMOTO
φ"(K)=0. Since every factor module of Ker(/) belongs to F(tQ/κ), Ker(/)
always belongs to F(tQ) by Lemma 2.1. On the other hand, f(φ"(K))=g(π(K))
=0 and φ"(K) C Ker (/)." Hence φ'\K) e T(tQ) Π F(tQ)=0.
Combining Theorem 3.2 with this proposition, we have
Corollary 3.4 ([3, Corollary 2.4]). For α quαsί-corαtίonαlly complete module
M having a quasi-projective cover, Hom
Λ
(M, —) is right exact on all exact
sequences of the form 0->L->M->iV->0 which is corational by N.
We do not know if this property of Homi?(M, —) characterize the quasi-
corationally completeness of M conversely.
4. Modules having projective covers
Let 0-*UΓ->P->Λί-»0 be a projective cover of a module M, i.e., it is an
exact sequence with P projective and K small in P. Throughout this section
we assume that M always has a projective cover and denote it by 0-+K-+P-*
M-+0. Wu and Jans ([9, Proposition 2.6]) has shown how to construct a
quasi-projective cover out of a projective cover. They showed that P modulo
the unique maximal i?-End#(P)-submodule contained in K is the desired quasi-
projective module. Here we shall claim that this submodule is precisely P(0: M),
where (0: M) denotes the right annibilator of M in R.
Proposition 4.1. The unique maximal R-ΈnάR{P)-submodule of P contained
inKisP(0:M).
Proof. First we shall show that P(0: M) coincides with Π{Ker(/):
/GHom
Λ
(P, M)}. This is a result due to Azumaya ([1, Proposition 7]). But,
for the sake of completeness, we give here its proof. Since P is projective, as
is well-known, there exist homomorphisms/
λ
eHomi?(P, R) and elements « λ € ?
for λGΛ such that, for each # e P , /
λ
(a?)=0 for almost all λ ^ Λ and x=
Σ*λ/λ(#). Put A= Π {Ker(/):/eHom^(P, M)}, and take x'(=A and M E M .
Then, for each λ, the mapping P-^M defined by x-*uf
λ
(x) for X E P is a
homomorphism and so uf
λ
(x')=0. Therefore f
λ
(o(/)^(0: M) and thus we have
χ*=J] ^ / X ( / ) E P ( 0 : M). This shows that AdP (0: M) and, since the reverse
inclusion is clear, we have the desired equality.
Now suppose that X is a submodule of P contained in K and is End^(P)-
allowable. Then, for each g: P->M, there exists /: P-+P such that af=g.
Take # e X Since f(x)&X, g(x)=a(f(x))=0. Thus we have x<=A=P(0: M).
Since P(0: M) is clearly a submodule of P contained in K and is Endi?(P)-
allowable, this completes the proof of the proposition.
We denote K/P(0: M) and P/P(0: M) simply by K and P respectively.
Then we have
A NOTE ON QUASI-CORATIONAL EXTENSIONS 691
Corollary 4.2 ([5, Corollary 3.4]). 0->K-*P->M->0 is the quasi-pro-
jectiυe cover of M, where & is the map induced by a.
Corollary 4.3 ([6, Theorem 2.3]). M is quasi-projective if and only if
K=P(0: M).
Proof. Since K is small in P, by [9, Proposition 2.1 and 2.2], M (
is quasi-projective if and only if K is End^(P)-allowable. Hence by Proposition
4.1 this is so if and only if K=P(0: M).
Now we shall show that the quasi-corationally completeness of M can be
described by means of the radical tP, Before we do this, we claim the following
Lemma 4.4. tP(R)=(tP(K)+P(0: M))/P(0: M).
Proof. To show this it is enough to note that, for each/: P->R, by the
projectivity of P there exists/: P^>K such that the diagram
P
Ί
K R
π
is commutative, and that conversely for each/: P->K the mapping/: P-*R
defined by fπ(x) = πf(x) for #e=P is a well-defined homomorphism since
Ker(τr)=P(0: M) is End*(P)-allowable. _
Using this lemma we see that P/tp(R) is isomorphic to PI(tP(K)+P(0: M))
and so by Corollary 2.5 we have
Corollary 4.5 ([3, Theorem 3.4]). The exact sequence P!(tP(K)+P(0: M))
->M->0 is maximal quasi-coratίonal by M.
As another application of the lemma we have
Theorem 4.6. M is quasi-corationally complete if and only if K=tP(K)-\-
P(0: M).
Proof. By Theorem 3.2 M is quasi-corationally complete if and only if
R=tP(R). Hence the theorem follows from the lemma.
Recently Bican [2, Theorem 4] has shown that M is corationally complete
(in the sense that it has no proper corational extensions by M) if and only if
K—tP(K). Hence by Theorem 4.6 for a module having a projective cover we
see that the corationally completeness implies the quasi-corationally com-
pleteness. Concerning the reverse implication, again by Theorem 4.6, we see
that a quasi-corationally complete module M is corationally complete if and only
if P(0: M)atP{K). Examples of a module which satisfies this condition are
692 Y. KURATA AND S. MORIMOTO
provided by a faithful module having a projective cover and any module having
a projectπe cover over a commutathe ring (cf. [3, p. 158 and Corollary 3.5]).
Finally in closing this note we give an example of a module which is quasi-
corationally complete but not quasi-projective.
EXAMPLE 4.7. Let N be the Jacobson radical (Φθ) of R and let N' (φO)
be a right subideal of N but not a left subideal of N. Then, since N' is not
Endfl(i?)-allowable, RjN' is not quasi-projective. RjN' has a projective cover
0-*N'-*>R-*>R/N'-+Q and, since T{tR) = mod-R, it is in fact (quasi-)cora-
tionally complete.
YAMAGUCHI UNIVERSITY
HAGI KOEN HIGH SCHOOL
References
[1] G. Azumaya: M-injective modules and M-projective modules, Preprint.
[2] L. Bican: Corational extensions and pseudo-projective modules, Acta Math. Acad.
Sci. Hungar. 28 (1976), 5-11.
[3] P.E. Bland: Quasi-corational extensions, Arch. Math. 23 (1972), 154-160.
[4] R.C. Courter: The maximal corational extension by a module, Canad. J. Math. 18
(1966), 953-962.
[5] K.R. Fuller: On direct representations of quasi-injectives and quasi-projectives,
Arch. Math. 20 (1969), 495-502.
[6] K.R. Fuller and D.A. Hill: On quasi-projective modules via relative projectivity,
Arch. Math. 21 (1970), 369-373.
[7] H. Katayama: On the cotorsion radicals, J. Fac. Lib. Arts Yamaguchi Univ. 8
(1974), 239-243.
[8] B. Stenstrόm: Rings of quotients, Springer-Verlag, Berlin-Heidelberg-New
York, 1975.
[9] L.E.T. Wu and J.P. Jans: On quasi-projectives, Illinois J. Math. 11 (1967), 439-
448.


Osaka University Knowledge Archive

http://dlisv03.media.osaka-cu.ac.jp/infolib/user_contents/sugaku/111F0000002-01503-18.pdf

A note on quasicorational extensions

Abstract

Similar works

Full text

Available Versions

Institutional Repositories DataBase (IRDB)

Osaka City University Repository

Osaka University Knowledge Archive