PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 48, Number 2, April 1975
RINGS OF EQUIVALENT DOMINANTAND CODOMINANT
DIMENSIONS
GARY L. EERKES
ABSTRACT.
sion
of rings
Continuing
of equivalent
dominant
tion
is teduced
can
be used
projective
These
as test
are
the proof
R\\,
being
noted
isomorphic
class
R-module
resolution
• • • , rz.
codom
Similarly,
and all modules
cogenerator
of a direct
ß-modules.
con-
for RM by
sum of a com-
For a left perfect
the codominant
than or equal
that
dimesnion
dimension
dimension
dimension
ring
of RM, de-
to n if for a minimal
is given
R
to be infScodom
by infidom
module
pro-
is
defined
dimRP:
for Ri (Rj)
in a dual manner,
Further,
dimRE:
„P
i = 1,
RE
for R left
injectivej.
projective!.
Finally,
provided
codom dimR3 (dorn dim„P = dorn dimRf).
dorn dimRJ
to the Society,
of
of RM is defined
of R is dorn dim„i?.
dim„â
is a test
codom dim_E=
Presented
hull
left
codominant
codom
RE iRP)
In [2] we showed
to
to be
dorn dim„J
we say that
for
is equal
left
The dominant
we define
is given
modules
for
and the left dominant
perfect,
modules.
projective
••• —' P nil1—>••• —►P, —>M —>0, P . is injective
The
dim „U.
injective
to an injective
to be greater
version
injective
ring with unity
of simple
11
of the
modules.
a minimal
RM we define
codom dim^M,
jective
1
2,
We denote
representative
R and a left
of the projective
ques-
dimension
the injective
a new and shortened
of the
the
cogenerator
of the injective
by when
dimension
R is an associative
are unital.
such
plete
Also,
injective
dimension
characterized
for rings
Specifically,
minimal
for the dominant
and the codominant
dimension
Throughout,
left
dimen-
is given
dimensions.
and the
the codominant
of the codominant
characterization
respectively
are in turn
2-injective.
that
the dominant
sidered
RR
modules
modules
examination
a categorical
and codominant
to when
conditions
modules
an earlier
and modules,
January
= codom dim„â.
25, 1973;
As an application
received
by the editors
it can
April
16,
1973.
AMS (MOS) subject
classifications
(1970).
Primary 16A60; Secondary
16A50,
16A52.
Key words and phrases.
Perfect
rings,
and dominant
dimensions,
2 -injectivity.
injective
projective
modules,
codominant
Copyright © 1975. American Mathematical Society
297
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298
G. L. EERKES
be shown
that the left codominant
lent
to the ring's
sult
that the minimal
„R
can be used
we propose
left dominant
injective
as test
to give
dimension
of a left
artinian
dimension.
In fact,
cogenerator
RU and the left regular
modules
for R9 and
a categorical
this
ring is equiva-
RJ,
characterization
follows
from the re-
module
respectively.
In this paper
of those
for which
rings
this can be done.
As shall
jective
be seen,
injective
annihilators
this
modules
(see
[3]).
is closely
related
is injective
to when the direct
and rings
To facilitate
this
sum of pro-
with ascending
discussion
condition
we provide
on
the following
definitions.
An injective
for all 0.
sider
module
Faith
0 of countable
ical dual
concept
For a left
RM is ¿^.-injective
[3, Proposition
3.3]
cardinality
we have
R-module
provided
has
shown
in testing
©XaM
that
is injective
it is sufficient
for S-injectivity.
to con-
As the categor-
n-projectivity.
RM and a subset
X C M we set
AnnR(X) = \r £ R: rX = 0!.
The collection
of all such
For the special
case
left ideals
of RM = RR,
for a given
QnnRÍR)
RM is denoted
is called
by QnnRiM).
the set of left annulets
of R.
1. The left regular
determine
when
dorn dim„R
tive
RR as a test
module.
module.
Clearly
this
In this section
is a triviality
we
when
= 0.
Theorem
mension.
module
RR is a test
1.1.
Let
ThenRR
modules
R be a left perfect
is a test
if and only
module
if each
ring
of positive
for the dominant
injective
left dominant
dimension
projective
di-
of the projec-
left module
is
¿.-injec-
tive.
Proof.
(=») It suffices
to show
that
©^ßPa
is injective,
Re for each
a = Q, is an indecomposable
injective
may assume
that
SQ.
Let
of O is
0—» ©SQPa—>£(©SnPa)
is injective,
for each
the cardinality
finite
Ei®^iiPa)
is isomorphic
of Q, Pa^Re„
©^ij'Pa
subset
0
Sa(BlrReß,
to a direct
of Û.
where
to a direct
Moreover,
each
summand
for infinitely
projective.
be the canonical
is isomorphic
many
of
Reß
0XpRe
ß £ V.
E(©SPa)
is injective.
„ for each
Thus,
©XßPa
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Pa =
Further,
injection.
summand
with
where
we
Since
Pa
of E{&2.QPa)
projective,
Whereas
finite
©20'Pa
subset
is injective.
Q,
RINGS OF DOMINANT AND CODOMINANT DIMENSIONS
(<=) For
injective
i
RP projective
resolution
we have
„P S ©Xn/<ea.
299
We consider
0—> Ren—>E,
—>E n „—»■■•
for each
a
l „—»•••
a.
a
a minimal
a=
fi.
Then, '
with dom dim„R > n,
a
provides
the desired
a
portion
Note that the sufficiency
dimension
sion
and
[3],
being
module
In this setting
R satisfies
a.c.c.
for
resolution
depended
of
K&2.Rea.
only on the dominant
nonzero.
// R is a left perfect
RR is a test
Proof.
of [3],
1.2.
injective
part of the proof
of the projectives
Corollary
ii
of a minimal
RJ,
ring
then
of positive
left dominant
dimen-
R is semiprimary.
E{RR) is S-injective.
on left annulets.
Then, by Corollary
But then
by Proposition
3.4
4.1 of
R is semiprimary.
Corollary
1.3.
A ring
R is left self-injective,
left perfect
and has
RR
as a test module for RJ if and only if R is QF.
Proof.
For such
a ring every
projective
left module
is injective
and so,
the ring is QF (see [3]).
In [3] Faith
S-injectivity
where
this
points
Proposition
1.4.
Then,
Proof.
pa:
Let
on left annulets
the converse
R be a left perfect
(lnnR{E{RR))
Let M be a subset
E{RR)—>Rea
= \r £R: r{\JaMa)=
Corollary 1.5.
lowing
a.c.c.
although
of R does
hold.
not imply
We provide
a case
ring
of positive
left dominant
= (innR{R).
of EiRR) = ®2üRe
is the canonical
a. Set Ma= pa(M)C Rea,
projection.
Then,
\r £ R:
rM = 0|
0\.
Let R be left perfect and dorn àtmRR / 0. Then the fol-
are equivalent:
(1)
RR is a test module for RJ;
(2)
R satisfies
(3)
For each
ideal
does
is true.
dimension.
where
out that
of E(RR),
I having
Proof.
2.
In light
in which
on left annulets;
ideal
the same
The minimal
that rings
as noted
a.c.c.
left
I there
right
of (1.4) this
injective
codom
in the previous
corresponds
annihilalor
as
is a restatement
cogenerator
dimR3 / 0 have
a finitely
generated
sub-
I.
of Proposition
as a test
module.
the same
S-injectivity
section.
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3.3 of [3].
First
we note
property
300
G. L. EERKES
Proposition
0.
Then,
2.1.
each
Proof.
Let
injective
Theorem
sequently,
this
R be a left perfect
projective
left module
3.1 of [2] states
is immediate
In establishing
ring for which
that
theorem
à'imJp = dom dimRJ
following
we shall
dimR3 /
¿.-injective.
codom
from the remark
our next
is
codom
make
. Con-
(1.1).
use of the following
ob-
servation.
Proposition
jective
fect
which
the converse
Proof.
index
set
0,
set
Re,
A.
Suppose
that
©SrRea
has
is injective
injective
is an injective
For
pro-
R per-
projective
a direct
every
only
summand
a direct
injective
simple
unique
simple
submodule.
has
projective
submodule
Hence,
IIaRe
For
&¿-¿Rea
fot
Rea's
isomorphic
are iso-
isomorphic
to ©£"
©2aRe
with
socle.
R-module
of ünRe
to Re.
fi infinite.
We again
let
and consider
is isomorphic
5r ©Xj-Re
lRe.
ate isomorphic
a nonzero
left
infinite
R-module.
to
a submodule
Rea's
of the form
module
fi infinite.
has
many of the
summand
Re be an indecomposable
left
many of the
has
UaRe,
Every
finitely
IIrRe
nonzero
for some
and so, isomorphic
The module
and so infinitely
R perfect,
RP
¿\-injective.
©X0Re
is projective
n in all.
so
©Li?ea
For
to show that
say
is
holds.
T, IIpRe
is impossible
Thus,
RP
then
Re an indecomposable
to Re,
©£"
This
ring and
also
set
some infinite
morphic
If R is a left perfect
is Tl-projective,
It suffices
any infinite
to
2.2.
module
and lIqR6
to Re's
is projec-
tive.
So as to relate
erator
the codominant
„LI to the codominant
dimension
dimension
of the minimal
of an arbitrary
injective
injective
cogen-
module,
we
shall apply the next result.
Proposition
2.3.
Let
R be left perfect
with
codom
dim„i! / 0 and
codom dim K„ll —
> ?2. Then,' ifi ■■—>P n —►...—>P IK—*„U —>0 is a minimal
projective
resolution
term in the minimal
of R U, P.,
projective
1 < i < n, generates
resolution
the
of an arbitrary
corresponding
injective
left
ith
R-
module.
Proof.
sis of (1.2)
By Theorem 3.1 of [2], (1.1) and (2.1), we note that the hypotheis satisfied.
Thus,
I
,£^E
R is perfect.
Rei\
®£_i
°i'V
For each
I
© ••• ©E
injective
Re \
©£
\
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°m/e»
RE,
RINGS OF DOMINANT AND CODOMINANT DIMENSIONS
where
e ,, • • • , e
shall establish
First
to a basic
set of primitive
(2.3) for E(©SftlS),
observe
nniF(S)minimal
belong
that
E(©SS)
idempotents
resolution
for R.
We
where S &Rex/}ey
is isomorphic
to a direct
summand
Next, let P lr¡—»., •■—»P —>E{S) —»0 be the first
projective
301
of
n terms in a
of E{S).
By (2.1) and (2.2),
Upln-*y-*UPn^UE(s)-*o
iîj
are the first
n terms
term in a minimal
rect
summand
©SpRe„
Sj
in a projective
projective
of Üjj.Pj
with each
and so P.,
„ll
is a test
Theorem
2.4.
mension.
., 1 < i < n.
®2rRe
(2.1) and (2.3)
we have the following
erator
resolution
resolution
of IlE(S).
of E(©ZßS)
Also,
Reg isomorphic
generates
Combining
Oj
such
Moreover,
a summand
to a direct
the zth
is isomorphic
summand
to a di-
is of the form
of P1;..
Thus,
P1¿,
ß.
along
with the results
characterization
of the previous
of when the minimal
section
injective
cogen-
module.
Let
R be left
Then the following
perfect
and of positive
left
codominant
di
are equivalent:
(1) RU is a test module for RS.
(2)
Each
injective
(3)
Each
injective
(4)
R has
a.c.c.
projective
left module
is
projective
left module
is l\-projective
on left annulets
and
E{RR)
¿.-injective.
is projective.
(5) RR is a test module for RJ and dom dim„R / 0.
Proof. (l)~(2).
This is (2.1) and (2.3).
(1), (2) «(3).
By Theorem 3.1 of [2], (1.1) and (1.2), we have sufficiency.
Necessity
is (2.2).
<=>(4). This
equivalency
is given
by Theorem
3.1 of 12], (1.4)
and Prop-
osition 3.3 of 13j.
«=»(5). See (1.1).
3.
restate
When codominant
the main result
a greatly
clarified
Theorem
3.1.
and dominant
dimension
of [2], replacing
are equivalent.
the long tedious
First
proof given
we
there
by
version.
Let
R be left perfect.
Then
codom dim „a > n if and only
if dorn dimRJ > n.
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302
G. L. EERKES
Proof.
(=») We induct
on n.
Let
„P
be projective,
and suppose
that
codom dimR£l > k + 1 and that dorn ditnRf > k. Consider
Pk + l
0_p^Ei_...^
Ek+i
l
0
where
the row is a minimal
injective
resolution
mal projective
resolution
of E,+1.
exact
whenever
RM is injective
functors
diagram results
columns
in the commutative
are exact and the first
To show that
£¿+1
HomR(Efe + 1, Ffc + j)j
module
of P..
Using
) has
since
Figure
M) and HomR(/V,
given by Figure
it will
suffice
a preimage
under
a case
and the column
RN is projective,
k + 1 rows are exact
in such
some fk+l = HomR(P, P,+.)
Hom„(_,
and
diagram
is projective
t £ HomR(E, + 1, E,+
Since
of RP
chase
and the last row semi-exact.
to show that
the identity
HomR(E^+1,
P A —>
t = Hom„(E,
as follows:
r
î
/2-/A-1
î
1
1. f xßk_{ / 0,
¿ = 1, •• • , k.
In this
are
1, in which all the
4+i -/a^o
Case
_)
the above
El + 1 will be isomorphic
1 we diagram
a mini-
case
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consider
map
to a sub-
+,, £,+.)
to
RINGS OF DOMINANT AND CODOMINANT DIMENSIONS
/fc+1
303
°
I
hl
+k+\b\~fk
°
î
î
h2
^kh2-fk-l
î
' 0
h
t
<b3bk_1-f2
o
î
hk
î
<f>2hk-fl
°
î
î
¿¿+i
Since (çij^ + i - i)^k = 0, (b.Y^k-r-^k.= ^V
Im t/Á fl Ker "¡^i^t+i
so h,+.
But if Ker<rSÂfe+i ^ °> then
^ 0> as Im "At ^s essential
is monic.
Thus,
E.+,
^A+ir«
in E,+..
is isomorphic
This
to a submodule
is impossible,
of Pj
and so,
is projective.
Case
2. fA¡>,_ . = 0 for some
we are back in the situation
(*=) This
is the dual
For those
rings
for Case 1 with
of the sufficiency
for which
and left projectives
are positive,
dominant
coincide.
dimension
1 < i' < k. Let
/ be the minimal such
i. Then
k = I - 1.
proof.
the dimensions,
respectively,
the theories
of codominant
of the injectives
dimension
and
Proposition 3.2. Let R be left perfect and dom dimRiP / 0. Then,
codom dimRll = dom dimRR.
Proof.
If not, then by (1.1), (2.1) and (2.4), codom dimRá = 0.
Corollary 3.3. // R is left perfect and dom dimRiP ¿ 0, then codom dim„U =
codom dim UR, where
Proof.
\AR is the minimal injective
As noted
orem 10], right
the following
Proposition
projective
ated
R must be semiprimary.
and left dominant
To throw slightly
make
in (1.2),
covers
cogenerator
more light
dimensions
on those
for \R.
But then
by L5, The-
are equal.
rings
for which
dom dimRJ
/= 0, we
observations.
3.4.
Let R be left perfect and dom dimRR > 0. Then, the
of injective
left modules
are direct
sums
injectives.
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of finitely
gener-
304
G. L. EERKES
Proof.
Rea
For a projective
is a submodule
cover
©SRea
of an injective
Re«.
—• _E —> 0,
we note
that
each
Considering
0
1
I
we note that
desired
©£Rea
is a direct
summand
of
©£Re¿¡
and, hence,
of the
form.
Corollary 3.5.
is finitely
Let R be left perfect and dom dimRR > 0. Then, if Rli
generated
injective,
or if the injective
indecomposable
projectives
are
X-
codom dim„ll ^ 0.
Proposition
an injective
3.6.
Let R be perfect and codom dimRll 4 0. Then, R has
projective
Proof.
Let
faithful
left ideal.
Re be an indecomposable
0 _> Re -
projective.
Then,
E(Soc Re) S E{Re)
©ZRe7j
ß
where
0
map.
of
is the projective
Since
cover
Re is finitely
©¿B'Reo
for some
generated,
finite
S of Re, there exists
ß
Thus,
paS:
the composition
is the natural
On the other
provides
hand,
morphic
©SßRe„
cover
a copy of each
Proposition
3.7.
codom
dim„ U > 0.
jective
left module
Proof.
= P®P
is
to a submodule
minimal
, where
and so there
R has
p:
E(Soc
the canonical
submodule
exists
by pcS.
of So'.
of pcß to P
Hence,
an injective
an injective
Re)—>E(S)
image
the restriction
is annihilated
left ideal
S is iso-
Re»
which
with
S
contains
of R.
R be left perfect,
dom dim„J
and dom
/ 0 if and only
dim„R
> 0 or
if each
injective
pro-
¿.-injective.
(=») If Re is injective
0 for infinite
P
left ideal
Let
is isomorphic
For a given
where
not annihilate
and
of P,
Then,
Soc (Re)
B Ç B.
with a monic
such that 5 = S ßi < Re a' and (ßiS rf) = S¿¡'.
Consequently,
minimal
can be completed
©S_Reo—*E(S),
does
to a submodule
as a submodule.
subset
£B
projection,
a projective
mapping,
and not S-injective,
Q.
(*-) See (2.4).
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then dom dim ©SflRe
=
RINGS OF DOMINANT AND CODOMINANT DIMENSIONS
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License or copyright restrictions may o
apply to redistribution; see http://www.ams.org/journal-terms-of-use
33
S
o
X
E
o
33
305
306
G. L. EERKES
REFERENCES
1.
H. Bass,
Finitistic
dimension
and a homological
generalization
primary rings, Trans. Amer. Math. Soc. 95 (I960), 466-488.
2.
G. Eerkes,
Codominant
Soc. 176 (1973), 125-H9.
3.
C. Faith,
Rings
27 (1966), 179-191.
4.
T. Kato,
579-584.
5.
Rings
dimension
of rings
of semi-
MR 28 #1212.
and modules,
Trans.
Amer.
Math.
MR 47 #3455.
with
ascending
condition
on annihilators,
Nagoya
Math.
J.
MR 33 #1328.
of dominant
dimension
> 1, Proc.
Japan
Acad.
44 (1968),
MR 38 #4525.
B. Müller,
Dominant
232 (1968), 173-179.
dimension
of semi-primary
rings,
J. Reine
Angew.
Math.
MR 38 #2175.
DEPARTMENT OF MATHEMATICS, VIRGINIA POLYTECHNIC INSTITUTE AND STATE
UNIVERSITY, BLACKSBURG, VIRGINIA 24060
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