BULL. AUSTRAL. MATH. SOC.
VOL. 13 ( 1 9 7 5 ) ,
I 6A2 I ,
08A I 5
349-353.
On semi-simple radical classes
B.J. Gardner and Patrick N. Stewart
It has been incorrectly asserted that each non-trivial semi-simple
radical class of associative rings is a variety defined by an
equation of the form
x
=x .
In this paper we give, for each
non-trivial semi-simple radical class of associative rings, a set
of equations which does define that class as a variety.
We shall discuss conditions on a class
are equivalent to
C of associative rings which
C being a semi-simple radical class.
See [7] for a
discussion of the more general case in which
C is a class of algebras
which are not necessarily associative rings.
Our theorem corrects an
assertion which appears to have been f i r s t made by Snider [6] and which has
been widely accepted by other authors.
If
a
is an element of a ring
which is generated by
for each
Let
p €P ,
a €R
R ,
[a]
denotes the subring of R
a , and the class of rings
is denoted by
R such that
[a] = [a]
B .
P be a finite non-empty set of prime numbers and, for each
N{p)
a finite non-empty set of positive integers.
(1)
The equations
(R{p : p € P})x = 0
and
(2)
pxIKaf
-1 : n € N(p)\ = 0 for each
p €P ,
where p = Tl{q £ P ; q ± p} , define a variety which we shall denote by
V(P, N) .
Received lU August 1975.
349
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350
B.J.
Gardner
and P a t r i c k
LEMMA. Each of the varieties
F is a field
q
in
V(P, N) , then
N.
Stewart
V(P, N) is contained
F is a subfield
in
8
of the field
}
and if
of order
for some q € P and m € N{q) .
Proof. Let a be an element of a ring R 6 l/(P, N) . Since the
g r e a t e s t common divisor of the numbers p , p € P , i s 1 , the equations in
(2) force
a € [a]2 .
Thus,
[a] = [a]2
and so
l/(P, N) c 8 .
Assume that F i s a field in f(P, N) . From equation ( l ) we see
t h a t F has c h a r a c t e r i s t i c q for some q £ P . Now, from ( 2 ) ,
nja' 7 ~ 1 -1 : n Z N(q)\ = 0
for each non-zero
a € F , so F i s finite. Let w € F be such that
m
[w] = F . Then ifi
= 1 for some m € tf(<?) and so F is a subfield of
the field of order
q™ .
II
THEOREM. Let C # {0} be a class of associative rings which is not
the class of all associative rings. The following are equivalent:
(i)
(ii)
(Hi)
Proof.
ring in B
C is a semi-simple radical
class;
C = l/(P, N) for some P and N as above;
C is a variety contained in
8 .
We shall use two results from [7]: Every finitely generated
is isomorphic to a finite direct product of finite fields
(Theorem 3-^); C i s
non-empty finite set
and such that a ring
generated subring of
in F (Theorem k.3).
a
F
R
R
semi-simple radical class if and only if there is a
of finite fields, closed under taking subfields
belongs to C if and only if every finitely
i s isomorphic to a finite direct product of fields
(i) °* (ii).
Let C be a semi-simple radical class, F the set of
fields whose existence is guaranteed by [7, Theorem U.3],
P = {p : 3 a field in F of characteristic p} and, for each p Z P ,
N(p) = {n : 3 a field of order
C = l/(P, N) .
pn
in
F} .
We will show that
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Semi-simple
It is clear that
radical
classes
C c l/(P, N) because if
35 1
0 t a $ R Z C , then
i s isomorphic to a finite direct product of fields in
Conversely, suppose
generated subring of
R € l/(P, A?) and
R .
5
[a]
F c l/(P, N) .
is a (non-zero) finitely-
By the lemma above,
R belongs to
B. , so
i s isomorphic to a direct product of finite fields [7, Theorem 3-4].
of these fields must be in
a subfield of a field in
5
f(P, N)
F .
5
Each
and so, using the lemma again, each is
Since
F is closed under taking subfields,
i s isomorphic to a finite direct product of fields in
F .
Thus,
R
is
C by [7, Theorem It.3].
in
(ii)
°* (Hi).
(iii)
"* (i).
finite fields in
[7,
This is the f i r s t assertion in the lemma.
Let
C.
Theorem 3.4] that
C be a variety,
Since
Cc 8
.
if
is in
F/ 0 .
Also, since
C is a variety,
C if and only if every finitely generated subring of
Theorem U.3] i t is sufficient to prove that
isomorphic fields).
Suppose
each
€P
u
[u] , the subring of
u = [u , u ,
HF
, u , ...)
\u "\ = P
such that
such that
u
R
In view of [7,
•
are fields in
Since
F
F .
For
€ Cc B ,
which is generated by
, is isomorphic to a finite direct product of
Thus, there are only a finite number of
possibilities for the characteristic of
k
F .
F is finite (we identify
F , F , ..., F , ...
finite fields [7, Theorem 3.4].
integer
F i s closed
again, we see that a
is isomorphic to a finite direct product of fields in
n , choose
F be the class of
C contains a non-zero ring, i t follows from
under taking subfields and, using [7, Theorem 3-^~\
ring
Let
=u .
F
.
Moreover, there exists an
It follows that there are only a finite
number of possibilities for the dimension of
F
over i t s prime subfield.
Hence there are only a finite number of possibilities for the fields
F., F , . . . , F , . . .
the theorem.
=x .
F is finite.
This completes the proof of
//
For each integer
x
and so
n 2 2 , let
V
denote the variety defined by
It follows from the implication (iii)
simple radical class.
°* (i) that
V
is a semi-
It has been claimed that every semi-simple radical
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352
B . J . Gardner
c l a s s i s one of the v a r i e t i e s
not correct:
each
V
and Patrick
V
N.
Stewart
(for references see [ 7 ] ) , but t h i s i s
contains t h e field of order
not contain the f i e l d of order
2 unless
2 , t u t f(P, N) does
2 €P .
Various other conditions are equivalent to those given in the theorem.
(iv)
(v)
(vi)
C is a homomorphieally closed semi-simple
class
(see [9, Corollary 32.2] for (i) *=>• (iv));
C is an idempotent (that i s , extension closed) variety
(see [9, Theorem 3^.11 for (iv) "=* (v) ) ;
C is a variety
with attainable
Theorem 1.5] for (i) <=* (iv)
(vii)
(viii)
identities
<=>
(see [7,
(vi));
C is a variety generated by a finite
fields (see [3] for (v) ~* (vii));
set of finite
C is a variety consisting entirely of arithmetic rings
(see [2] for (i) *=> (v) <=> (viii) and [4] for
(vii) <=> (viii) ) .
Finally, we note that other equational definitions of these varieties
are considered in [5] and [8].
References
[7]
B.J. Gardner, "Semi-simple radical classes of algebras and
a t t a i n a b i l i t y of i d e n t i t i e s " , submitted.
[2]
L.C.A. van Leeuwen, "A note on primitive classes of r i n g s " , submitted.
[3]
/I.M. MapTbiHOB [L.M. Martynov], -"0 paapewmux HO/vbijax" [On solvable
r i n g s ] , Ural. Gos. Univ. Mat. Zap. 8 (1972), t e t r a d ' 3, 82-93.
[4]
Gerhard Michler und Rudolf WiIle, "Die primitiven Klassen arithmetis
arithmetischer Ringe", Math. Z. 113 (1970), 369-372.
[5]
/l.H. WeBpMH, /I.M. FlapTbiHOB [L.N. Sevrin, L.M. Martynov], "0 flocTMWMMbix
K/iaccax a/ire6p" [On attainable classes of a l g e b r a s ] , Sibirsk.
Mat. 2. 12 (1971), 1363-1381.
[6]
Robert L. Snider, "Complemented hereditary r a d i c a l s " , Bull.
Math. Soc. 4 (1971), 307-320.
Austral.
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available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700024606
Semi-simple
[7]
radical
classes
Patrick N. Stewart, "Semi-simple radical classes", Pacific
353
J. Math. 32
(1970), 21+9-25**.
[S]
Heinrich Werner und Rudolf Wi Me, "Charakterisierungen der primitiven
Klassen arithmetischer Ringe", Math. Z. 115 (1970), 197-200.
[9]
Richard Wiegandt, Radical and semisimple classes of rings (Queen's
Papers in Pure and Applied Mathematics, 37. Queen's University,
Kingston, Ontario, 197*0 .
Department of Mathematics,
University of Tasmania,
Hobart, .
Tasmania;
Department of Mathematics,
Dalhousie U n i v e r s i t y ,
Ha I i fax,
Nova Scotia,
Canada.
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