I n. J. Math. Math. S ci.
Vol. 5 No. 4 (1982) 763-777
763
THE KRULL RADICAL, k-PRIMITIVE RINGS,
AND CRITICAL RINGS
RALPH TUCCI
Department of Mathematics, University of New Orleans
New Orleans, Louisiana 70148 U.S.A.
(Received October 21, 1981 and in revised form June 15, 1982)
ABSTRACT.
We generalize results on the Krull radical, k-primitive rings, and critical
rings from rings with identity to rings which do not necessarily contain identity..
KEY WORDS AND PHRASES. Krull radical, prime radical, Jacobson radical, Kru11 dimension, noetherian, k-primitive, critical, co-critical.
ANS 1980 SUBJECT CLASSIFICATIONS.
I.
INTRODUCTION.
Primary 16A33, 16A55; secondary 16A20, 16A21, 16A34
In this paper we extend some results on Krull dimension from rings
with identity to rings which do not necessarily contain identity.
to embed a ring R into the usual ring R
1
The basic idea is
with identity, and to study the relation
between the right ideals of R and of R
I.
In the first section of this paper we use Krull dimension to define the Krull
radical of R, denoted K(R).
The Krull radical is a generalization of the Jacobson
radlcal, and was first defined by Deshpande and Feller [i] for rings with identity.
Our main result in this section is that K(R)
K(R1).
This enables us to use previous
work i** [i] to characterize the Krull radical as the annihilator of all critical R-
modules, which in turn lets us determine the Krull radical of the n x n matrix ring
over R.
We then describe the relation between the Kruil radical of R and that of a
two-slded ideal I
R.
Finally we derive containment relations between the Krull
radical on the one hand and the Jacobson and prime radicals on the other.
In the next section we look at k-primitive rings, which are a generalization of
R. TUCCI
764
We list the main properties of these rings and generalize slightly
primitive rings.
a theorem on these rings (Prop. 3.4).
We finally turn our attention to crit-
ical rings, which are closely related to k-primitive rings.
Necessary and sufficient
conditions are given for a critical ring to be a domain, and those critical rings
which are not domains are completely characterized.
In what follows the letter R denotes an associative ring which does not necessarAn R-module
ily contain identity.
is a right R-module; usually we will simple call
this module M.
Let Z denote the integers.
We define R
(r, n) Ir
I
is componentwlse and multiplication is given by
nn’).
R, n
Z},
(r’, n’)
(r, n)
where addition
(rr’ + nr’ + n’r,
This is just the usual ring with identity in which R +/-s embedded.
For nota-
tional simplicity, we identify R with the subring (R, 0) of RI to which R is isomorphic.
an
any
mr
R-module M can be considered
I are unital, so that every
RI. Conversely,
mr + mz for all m
if we define m(r, z)
M, (r, z)
All modules over R
Rl-mOdule
Rl-modle M
can be considered an R module with scalar multiplication defined by
m(r, 0) for all m
M, r
R.
Krull dimension for an R-module M is defined as
in [2] and is denoted K dim M, or sometimes K dim MR
of this paper is assumed.
A familiarity with the results
Note that for any R-module M, K dim
is a k-critical R-module if and only if M is a k-critical
Rl-mOdule.
K dim
i
and M
Finally, E(M)
denotes the injective hull of this module M.
THE KRULL RADICAL.
2.
R
is a
As in [i] we say that a right ideal H of a ring R is k-co-crltical if
k-critical R-module.
0
n
A right ideal H of R is n_-modular if there exist e
Z, such that er
H for all r
nr
accordance with the usual terminology.
R.
If n
R,
I, then we call H modular in
A right ideal which is either maximal modular,
1-co-critical and n-modular, or k-co-critical, k > 2, is called a special co-crltical
r_ht ideal of R.
The Krull radical of R, denoted K(R), is defined to be the inter-
section of all the speical co-critical right ideal of
none, then we define K(R)
cides with that given in
R.
R, if any exist; if there are
Note that this definition of the Krull radical coin-
Ill if R has identity.
In order to be able to use the
765
THE KRULL RADICAL, k-PRIMITIVE RINGS, AND CRITICAL RINGS
K(RI).
results of [l], we first prove that K(R)
LEMMA 2.1
er- nr
H
E
Let H be a right ideal of R, H
R, and let H I
{(e, -n)
E
R
I
E}.
for all r
Then
(1)
H
(2)
H
the unique right ideal of R which
I is
l
that H n R
H;
I
is the n-modular if and only if H
(i)
PROOF
ness of H
R.
I
It is routine to verify that H is a right ideal of R
I
1.
that H
I
I follows from the observation
(2)
(e, -n)
This follows because H
R with n
1
LEMMA 2.2
I
{x
R
I
x R
R if and only if H
The unique-
__c H}.
I contains some
0.
0 for every m
Let M be a trivial R-module; i.e., mr
M, r
E
PROOF
Since M is a trivial R-module, its R-module
i.e., as a Z-module.
structure as an abelian group
struc{ure is the same as its
By [2, Cor. 4.4] K dim MZ <
I.
K dim Z z
K(R)
THEOREM 2.3
K(RI).
By [3, p. ii, Thm. 2] and by definition of the Krull radical,
PROOF
J(RI)
J(R)
R.
critical right ideals of R
Thus, K(R I)
I.
n
HIC
(H 1 n R), where C is the set of co-
We will show that the set of special co-crltical right
ideals of R coincides with the set of right ideals of the form H I n R, where H I
Since
R.
k exists, then k < I.
If K dim M
K(RI)
is maximal with respect to the property
J(RI)
J(R)
ideal of R
we do not need to consider the case where H
I is a maximal right
I.
Suppose that R has some special co-critical right ideals.
Let H be a special k-
co-critical right ideal of R, k > 0, and let H be as in Lemma 2.1.
1
R
mine K dim
C.
I
i"
By [2, Lemma i.i]
We first deter-
R. TUCCI
766
R
1
H
K
diml
R
I
K dlm
R + HI
H
sup
K dim
+ H1
HI
1
K dlm
sup
R
1
R+H
Now
+
K dim
H1
H1
R
is a homomorphic image of
1
R
1
R+H
K dim
R
R
-< K
1
1
dim--=
-,1 which is a trlvial R-module.
1 by lemma 2.2.
Since
R+HI
H
1
R
R o H
1
Thus,
R we have
H
To show. H 1 is co-critical, let K1 be any right ideal of R1 which properly conThen R o K properly contains H because of the way H1 is defined.
I
R
R
1
1
ing the argument we used to find K
gives us that K dim
tains H
R
R o K
K dim
< K dim
R
assume H
I
Let H
contrary to our assumption).
is k-critical by
[2, Prop. 2.3].
to
H
I
H
1
H
I.
H
_Clwe
HI
2 then H is a special co-critical right ideal
If k
i.
Then H
I
R; for
if H
I
and
then
then, that H 1 contains some (e, -n)
nr
then R has no special co-critical right ideals
R + HI
RI
have that
Since
R o H
1
R
H, contradicting the fact that K dim
R
0
I.
R, then there is an onto map from
R
I is critical, we must have
Since both modules have Krull dimension i,
I
RI
R
But
0, which implies R H 1 and this in turn implies that
of R. Suppose k
R
R
I
1
HR R-.
1
1
=-.
i
1
is a k-co-critical right ideal of R
is a k-co-critical right ideal of RI, k > 0, and
R (if there is no such H I
I
K
diml
Therefore H
k.
Conversely, suppose that HI
R
Repeat-
I.
0
R
H.
Hence H
R with n
+ 0,
i.
Thus, HI
so for every r
R.
We must have,
R, (e, -n)r
and therefore special.
I is n-modular,
R by
Suppose now that R has no special co-critical right ideals. Then K(R)
R
1
definition. Since-is 1-critical, R is a co-critical right ideal of RI. Every
other co-critical right ideal H of R contains R; for, if R
I
I
special co-critical right ideal of R, contradiction.
H
I
then H
Therefore, K(RI)
R is a
I
R
K(R).
er
THE KRULL RADICAL, k-PRIMITIVE RINGS, AND CRITICAL RINGS
767
This completes the proof.
COROLLARY 2.4
K(R) is the set of elements of R which annihilate every
(i)
critical right R-module.
K(R) is a two sided ideal of R.
R
0.
(2)
(3)
K(KR)
PROOF
[I, Thm. 2.1].
This follows from Thm. 2.3 and
The next result shows that
matrices over R.
K(Rn)
(K(R)) n, where
If R has identity, then
Rn
is the ring of n x n
denotes the matrix wlth i in the (i, j)
Eli
position and zeroes elsewhere.
LEMMA 2.5.
Let R be a ring with identity, and let H be a right ideal of R.
Take H (1) to be the set of all matrices in R
whose i
th
row has entries from H and
R
whose other entries are arbitrary.
Then
is a critical
(i)
-module
if and only if
H is a critical R-module
R
For simplicity, asse that i
PROOF
whose only non-zero row is the first.
Note that
i.
n
Htj
conslstsR
n
Thus, any smodule S of
H
S
0
0
0
0
R
NI,
where
Nn are subsets of R.
R
n
H(1) Ejj
n
k
any i
n,
can be itten
Nn;
I
for
R
H-
entries in the
Now N
(i)
of matrices
for any I
j
n, and
n
(l, j) position
i.e., from
R
R
n
n
.
1
N
and zeroes elsewhere.
This implies that
H(1) Ejj Ejk H(1)
arbitrary, we have that N
consists of matrices with nonzero
Ejj
Nn.
Call this set N.
Nj
N
k.
But then for
Since j and k are
It is routine to check that
N is an R-submodule of R
Thus, there is a i-i onto order preseing map f from the
R
R
n
/ N.
0
0
of
to the R-submodules of
The
given by f:
-submodules
IN’’" N1
H(1)
LO...oJ
result follows iediately from this.
LEMMA 2.6.
Let R be a ring with identity.
(+/-)
then there is a co-critical right ideal H
=
If M is a cyclic critical R-module,
Rn as in Lemma 2.5 such that M is
R. TUCCI
768
R
isomorphic to
H(i)
Since M is cyclic, we can find a matrix A
PROOF
first that there is an integer j, i
.th row.
X 2,
Rn such that AX
k
Xn
n(AXn)Rn
n
ARn
We show
< n, such that every element in M has non-zero
Suppose this is not the case.
3
(AXI)Rn
-< j
M such that M
Then there is a collection of elements
0 and the k
th
row of
k
XI,
But then
is zero.
contradicting the fact that M is a uniform module by [2
0
Cor. 2.5 and 2.6].
We can assume without loss of generality that every non-zero element of M has
Let M’ be the module consisting of all matrices whose first row
non-zero first row.
appears as the first row of a matrix in M, and whose other entries are zero.
M
a map f:
+
M’
If A
as follows:
M, then f (A) is the matrix whose first row is the
same as that of A, and whose other entries are zero.
isomorphism and
M’
is of the appropriate form
K(Rn)
THEOREM 2.7.
PROOF
R
n
H (i)
where
Eii
X
K(Rn)
Ejj
H(i)
Let X
R
a two-sided ideal of R.
so is
;
so that
thus, by Cor
n
This completes the proof.
Let M be any cyclic critical R -module.
n
H(i)
x
n
is defined as in Lemma 2.5.
K(Rn), x
Eij 0.
that x annihilates the critical R-module
R
This map is certainly an R
(K(R))n
First assume R has identity
Lemma 2.6, M
Define
2.4 (1), x
K(R)
By Cor. 2.4 (2), F(R) is
the (i,j) entry of X.
.
R
By
Then x
Ei
J
As in the proof of Lemma 25, this shows
Since M is an arbitrary cyclic
Therefore, K(Rn)
inclusion follows by reversing the steps of the argument.
(K(R)) n
Hence K(R
Rn module,
The reverse
n)
(K(R)) n when
R has identity
If R does not contain identity, embed R into R
I.
Thin. 2.2 we have
(K(R)) n
(K(RI))n
K((RI)n).
However, just as a critical module
over R can be considered a critical module over R
modules over
Rn and (Rl)n.
From the previous paragraph and
1
and vice versa, so we can identify
Therefore, by Cor. 2.4 (i),
K((RI)n)
K(Rn).
This
completes the proof.
We now describe the relation between the Krull radical of a ring R and that of a
THE KRULL RADICAL, k-PRIMITIVE RINGS, AND CRITICAL RINGS
769
two-sided ideal I in R.
K(R), let I be a two-sided ideal of R,
Let R be a ring such that R
LEMMA 2.8.
and let M be a k-critical 1-module.
Then either MI
0 or MI is a k-critical R-
module.
Assume MI
PROOF
CR
+ 0,
and take C to be a critical R-submodule of MI.
0 by Cor. 2.4 (1), so CI
K dim MI
R
> k.
0.
Hence K dim C
K dim C
R
Then
k, which implies that
I
Since the reverse inclusion always holds, we have K dim MI
k.
R
That
MI is a critical R-module follows from the fact that MI is a critical 1-module.
PROPOSITION 2.9.
Then K(1)
ideal of R.
I.
for which Mi
+
0.
Since MiR
Hence the map f:
MIR
M
But then for any m
contradiction.
Therefore MI
EXAMPLE 2.10.
Krull radical.
then there is some i
mi for all m
f(m)i
M, f(mi)
0, and by Cor. 2.4 (i), I
mi
2
M is actually an
0 and hence Mi
0,
K(1).
By [4, T. 48], this is equivalent to the fact that J(1)
x]]
F
F[
0
F[x]J
R
I
Prop. 2.9 is true if we substitute the Jacobson radical for the
for any ideal I of a ring R.
Let
+ 0,
0 by Lemma 2.8 and Cor. 2.4 (i), Mi is a critical
Mi defined by f(m)
1-isomorphism.
K(R), and let I be a two-sided
If MI
Let M be a critical right 1-module.
PROOF
R-module.
_-
Let R be a ring such that R
I n J(R)
Unfortunately, this does not hold for the Krull radical.
where F is any field, x is a couting indeteinate over F,
and the ring operations are the usual matrix addition and multiplication.
By [i,
0
gx. 4
K(R)
0.
then K(I)
However, if we take I
0
I because I has
0
no special co-critical right ideals; for, since I is isomorphic to a direct sum of
I
copies of F, any special co-critical right ideal would have to be maximal modular.
Certainly I has no such right ideal.
We now describe the containment relations between K(R) on the one hand and J(R)
and P(R) (the prime radical of R) on the other.
770
R. TUCCI
PROPOSITION 2.11.
(i)
For any ling R, K(R) S J(R).
(2)
If R is a ring with Krull dimension, then K(R)
(3)
If R is a commutative ring, then P(R)
If we embed R into
PROOF
RI,
c
__c P(R).
K(R).
P(RI),
then P(R)
J(RI), and.K(R)
J(R)
K(R I)
by [4, Cot. after Thm. 59], [3, p. ii Thm. 2], and Thm. 2.3 of this paper respectively.
Hence we may assume that R has identity.
Now (i) follows from the definitions of
K(R) and J(R), while (2) and (3) are mentioned in [i, p. 188] for rings with identity.
EXAMPLE 2.12.
The containments in Prop. 2.11 (i) and 2.11 (2) are both
Then K(R)
Let R be as in Ex. 2.10.
proper.
(2)
(i)
The containment in Prop. 2.11 (3) also is proper.
] where
Xn
indeterminates.
j
x2j+l x2j+2,
Then x
#
{xl,
x
}
xn,
2
Z2[Xl,
.
Take I to be the ideal generated by the polynomials
S
and let R
i, 2,
f
Say that
x2j_l x2j
I
0
0
y
Let
b--
a
S
x in R for all
Now b
P(R), but b
Z2
(I’ + I)
0
0
is the
]
is critical but
K(R), because a map f:
dimension, has kernel containing almost all the x.
3.
J.
where these symbols are
ideal of S generated by all xi, j > i, then C
+ 0.
x2j_l x2j +
K(R); for, if I’
Then a e P(R), but a
defined in the previous paragraph.
C a
x2,
K(R) by [5, Ex. 4.17].
P(R), but x
0
Let S
is a countably infinite set of commuting
In general, P(R) and K(R) are incomparable.
(3)
+ 0.
0, but P(R)
’s,
S
/
M, where M has Krull
and hence x.
CO-PRIMITIVE IDEALS
Just as J(R) can be expressed as the intersection of certain two-sided ideals of
R, so can K(R).
right ideals of
K dim
R
?.3
Let
RI,
HI,
H 2,
Hn be a
and suppose that
finite collection of special co-critical
E() E()
for all 1 < J, k < n. If
3
k for all i < j _< n, then the largest two-slded ideal D
a k-co__-primltive ideal of R.
n
c__ o
Hj is called
j=l
An ideal which is k-co-primitive for some ordinal k is
THE KRULL RADICAL, k-PRIMITIVE RINGS, AND CRITICAL RINGS
R
-< n}.
i < j
Here
{r
It fs not hard to see that D
called co-primitive.
(H’)I3
R
I
r
Hj) I
771
0 for all
is the extension of H. to a co-critical right ideal of R
I as
in Lemma 2.1.
THEORY4 3.1.
R,
From Cot. 2.4 (2), K(R) is a two-sided ideal of R.
PROOF
every special co-critical right ideal H
Thus, K(R)
ideal of R.
n D.
=
R, then K(R)
Conversely, if r
previous to this theorem we have M
n D
_ _
K(R) is the intersection of all the co-prlmitive right ideals of
r
0
Since
K(R)
H for
D for every co-primitive
D, then by the observation
0 for any critical R-module M.
Thus,
n D.
K(R) by Cot. 2.4 (i), so K(R)
If 0 is a k-co-primitive ideal of a ring R with Krull dimension k, then R is said
to be k_-primitive.
This definition coincides with that given in [6].
PROPOSITION 3.2.
Let R be a ring with Krull dimension k.
Then R is k-primitive
if and only if R has a faithful critical finitely generated module C with K dim R
K dim C.
Suppose that R is k-primitive.
PROOF
special k-co-critical right ideals
E
for all I
E
R
that each
1
(Hj)’
is critical by
K dim C.
j, k
n.
HI,
Then there is a finite collection of
Hn whose
But then
E((Hj)I
intersection is 0 and such that
E(
so we may assume
R
lles in the same injective hull.
The module C
1
(HI)
R
+
[6, Lemma 3.1], finitely generated, and faithful, and K dim R
+
1
(Hn) I
k
The converse follows by reversing the steps of this argument.
The main properties of k-primitive rings have been investigated in [6].
some of these properties here.
We list
Recall that the assassinator of a uniform module C
over a ring R with Krull dimension is that ideal P which is maximal among the
annihilators of submodules of C.
THEOREM 3.3.
Let R be a k-primitive ring with faithful critical module C, and
R. TUCCI
772
let P be the assassinator of C.
(1)
0 for two right ideals A and B, then either A
If A, B
0 or B
P (i.e., R is
P-primary)
(2)
P is the only prime ideal of R which is not a large right ideal;
(3)
if H is any non-zero right ideal of R, then K dim H
(4)
R and C are nonsingular;
(5)
if H is a large right ideal of R, then K
(6)
the injective hull of R is a simple artinian ring.
K dim R;
R <
K dim R;
dim
In [7, Thm. 3.4], Boyle, Deshpande and Feller characterize a k-primltive piecewise domain (PWD) which contains a faithful critical right ideal.
this type of ring as a BDF ring after the authors.)
scribe a slightly broader class of rings.
This result can be used to de-
Recall that a PWD R is a ring with identity
en such
which contains a complete set of orthogonal idempotents e l,
x
e
i
R
ej,
y
E
R
ej
O.
0 or y
0 implies x
then x y
ek,
(We shall refer to
that if
In what follows, we
assume that R is written as an n x n upper triangular matrix ring; see [8].
Recall
also that a ring S is a quotient ring of R if R is a large R-submodule of S.
In the next result, we assume that R is a noetherian k-primitive ring with
identity which is a direct sum .of non-isomorphic critical right ideals (and hence is a
PWD by [8]).
E(R) is a matrix ring over a division ring D with identity I, we
Since
can define the matrix M
position and
Ell
O’s elsewhere, i
PROPOSITION 3.4
+
+
< j < n.
Let R and M be as above.
_
is the
Eln where Eli
matrix
with i in the (l,j)
R + RMR
Then R has a quotient ring S
which is a noetherian BDF ring if and only if (RMR)
2
RMR + R and RMR is a finitely
generated R-module.
PROOF
Note that R is an upper triangular matrix ring with
Also, each e Re. is noetherian, for if I
J J
Finally, note that R
E
kj
ekR +
e
>j
j
Re
ejRek
then e.Re
3 J
0 for j > k.
R
Y"
elS.
Let S
(RMR)
2
R
+
R + RMR.
RMR.
Assume that RMR is a finitely generated R module and that
Since S
E(R), S
is a quotient ring of R.
Also, S is a finitely
THE KRULL RADICAL, k-PRIMITIVE RINGS, AND CRITICAL RINGS
773
generated R-module, which implies that S is a noetherian ring and that
Now
finitely generated R-module.
R-module by [9, Cor. 2.4].
dim(elS
of e Re
n n
K
dim(elS) R
S
elS
(---) S
elSej
K
dim(elS) R
elS
(.i)
(elS) R
for, since R is a PWD,
elSe n
is merely a sum of copies
< K dim
is noetherian.
elSen_ 1 + elSe n
elSen
because S’
S’
S
in this manner,
(elSen)enRen
K
shows that
Since S is a
0.
Conversely, let S
S’
Then
elS.
K dim
dim(elSen) R
0 for some s
s
ejse k
(elSen) S
is a critical
(3.1)
Further,
This together with
for, if
elS
(---) R
elS
is a
<
(elSen) S
Thus, K
which is uniform, so
H be an S-submodule of
Let 0
K dim
But K
elS _c E(elR
elS
R
dim(elS) S
elS
(3.2)
(elSen) R
so that K
dim(elS
is a critical S-module.
S, then for any idempotents
PWD,
+ RMR
ej
se
k
0.
(elSen) R
Since S
is faithful;
R we have
S is noetherian
Let
is noetherian.
Again, S’ S being noetherian implies
k
elS
dim(elS) S
0.
Therefore, s
be a noetherian BDF ring.
But by (3.2),
e
ej,
K
R
Now
S’R is
noetherian,
so that (elSen_ +
elSen) R is noetherian. Continuing
I
en-iRen-i S’R,
we have
elS R noetherian, and hence R is finitely generated.
Finally, since S is a ring, (RMR)
2
__c R + RMR.
Prop. 3.4 applies more generally to a ring R with identity which is a direct sum
of non-singular non-isomorphic critical right ideals; such a ring is a direct sum of
ideals, each of which is a k-primitive ring by [i0, Prop. 5.3].
EXAMPLE 3.5.
R
(i)
F
0
F[x]
0
F
Fix]
0
0
Fix]
Let F be a field, x a commuting indeterminate over F, and let
with the usual matrix operations.
Then R satisfies the
R. TUCCI
774
conditions of Prop. 3.4.
RMR
(2)
0
0
0
0
0
0
If
then S
R
+ RMR
0
F
Fix]
0
0
Fix]
is a BDF ring.
Let x and y be commuting indeterminates over F, and let
R
Then S
F[x]
0
0
F[y]
0
0
F[x]
0
Fix, yJ F[x, y]
F[x, y]
F[y
0
0
F[x, y]
F[x, y]
Fix, y]
and RMR
IF[x]
Fix, y]
Fix,
0
0
0
0
0
0
Y]I
has no Krull dimension, since Fix,
y] does
not
Fix, y]
have finite uniform dimension as an F[y] module.
In this case RMR is not finitely
generated.
4.
CRITICAL RINGS
A ring R is critical if
If R has identity, then
is a critical R-module.
faithful, and hence R is k-primitive.
is
Thus, we could describe the structure of this
However, it is possible to prove more about R, even if we do not
ring using Thm. 3.3.
assume that R has identity.
PROPOSITION 4.1.
PROOF
given by f(r)
If R is a domain with Krull dimension, then R is critical.
Let C be a critical right ideal of R, 0
c E C.
The map f:
R
C
cr is i-i, proving R is critical.
The converse of Prop. 4.1 is true if R has identity.
To examine this converse
for k-critical rings which do not possess identity, we need to consider separately the
cases k > 1 and k
N
Recall that a module M is monoform if, for any submodule
i.
M, a homomorphism f:
N
M is either zero or i-I.
Any critical module is mono-
form by [2, Cor. 2.5].
PROPOSITION 4.2.
If R is a k-critlcal ring, k > i, then R is a domain.
THE KRULL RADICAL, k-PRIMITIVE RINGS, AND CRITICAL RINGS
{r
Let T
PROOF
Hence T
contradiction.
0}.
By Lemma 2.2, K dim T
R
rR
0.
Now if a, b
because R is monoform, so a
R with ab
1 < K dim R,
0, then either b
a
{ak
P
THEOREM 4.3.
P
Q, p a fixed prime}, and Z
k
Let R be a 1-critical ring.
(i)
R is a domain;
(2)
R
(3)
0 is an n-modular right ideal of R.
2
=>
R
2
(3)
Assume R
2
Then the following are equivalent:
xR
If there is 0
R such that xr
Hence xZ o xR
is artinian.
nx.
n
Z, 0
R such that nr
r
0 for any 0
r
R.
0, then
Otherwise, since
Now the module
being a proper homomorphic image of the 1-critical module xR
xZ n xR
x e
0.
R because R is monoform and so 0 is n-modular.
0, we can pick x
xR + xZ
xZ
0
G(p)
Z
p
(2) Trivial.
(i)
0 for all r
nr
Q is the set of
+ 0;
PROOF
(2)
0 or a e T
0.
To examine l-critical rings, we need the following notation:
rational numbers, G(p)
775
R, n
In particular, there exist e
0.
Multiply on the right by any r
+ xZ,
Z such that
R and cancel the element x to show
that 0 is n-modular.
(3)
In this part of the proof we use the argument from Ill, Prop. 4.1].
(i)
be an n-modular ideal of R; i.e., there are e
R.
all r
nr
{r
Let T
0 for some 0
r
R
0}.
r R
R, then nR
0 for any 0
with ab
T.
If b
implies that aR
r
R.
0 and so a
0.
T.
Z such that er
O, so that T
Hence t
R
Since abR
Hence
R
O.
is a domain; for
O, the fact that R
is a domain.
0 for
nr
0.
If
T generates a
0; in particular, any element t
We note first that
T, then bR
n
As in Prop. 4.2, we show that T
finite, hence artinian, right ideal of R.
that nr
R, 0
Let 0
Now suppose
let a, b
is monoform
Because R is 1-critical, R is
an artinian domain, and hence is a division ring D.
Define a group homomorphism f:
T is fixed but arbitrary.
0
t
r
T, then rR
R
/
T by f(r + T)
rt for all r
This map is i-i; for if rt
0 because R is monoform, and hence r
R
R, where
0 for some r
T, contradiction.
R,
Hence T
R. TUCCI
776
Now R, and hence D, has no elements of finite
contains a subgroup isomorphic to D.
This implies that D, and hence
order by assumption.
isomorphic to Q.
T.
Without loss of generality we write Q
QR
module, which implies that K dim
It follows that T
contradiction.
T, has a subgroup which is
K dim
QZ"
Thus, Q is a trivial R-
However, K dim
0, and R is a domain.
QZ
does not exist,
This completes the proof.
The case when 0 is a maximal modular right ideal is handled similarly.
Hence we
may summarize:
COROLLARY 4.4.
Let R be a critical ring.
is a special co-critical right ideal of
is isomorphic to a finite sum of Z and
of
Zp’S
and
0.
0.
Then as a group R
G(p)’s for various primes p.
Since the injective hull of R is isomorphic to Q, identify R with some
subgroup of Q.
Thm. 9 24]
R; otherwise, R 2
Let R be a l-critical ring satisfying R 2
THEOREM 4.5.
PROOF
Then R is a domain if and only if 0
If G is finitely generated, then G is isomorphic to Z by [13,
If G is not finitely generated, then
Zpn’S,
b for some
G
+ Z
is isomorphic to a direct sum
where there is a distinct summand for every prime p which divides
G.
EXAMPLE 4.6.
Let R
__a
2k
a, k
Z} where the product of any
two elements is O.
Then R is l-critical but not right noetherian.
We note that Hein [12] has recently generalized Thm. 4.5.
ACKNOWLEDGEMENT
Much of this paper is taken from the author’s doctorial thesis.
The
author would like to thank his advisor, E. H. Feller, for his kindness, encouragement,
and boundless patience.
The author would also like to thank the referee for his help-
ful comments which have led to substantial improvements in this paper.
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