T h e proof will be c o m p l e t e d b y showing t h a t i) implies iii). I f A can be i m b e d d e d as a n /-ideal in a n / - r i n g w i t h identity, t h e n A is a n / D - r i n g (Corollary 4.5) a n d A satisfies condition (*) (Theorem 3.2). I f a, b ~ A + are such t h a t ab > b (or ba > b) then, b y condition (*), [(ab - b) V (ba - b)] A [(c -- ac) V (c -- ca)] <~ 0 for each c e A +. Hence, t h e e l e m e n t a of A satisfies (e - ac) V (e - ca) = e - ac A c a ~< 0 for each c e A + , contradicting the fact t h a t A is an/D-ring. Hence, iii) is satisfied.
F o r a r b i t r a r y /-rings, no such characterization is known. I n fact, it is n o t k n o w n w h e t h e r it is sufficient t o require o n l y t h a t A ' be a lattice u n d e r the strong order. I t is n o t inconceivable t h a t , if A ' is a lattice-ordered ring, t h e n t h e special n a t u r e of A ' a n d the presence in A ' of the m a x i m a l / - i d e a l A, which is itself a n / - r i n g , force A ' to be a n / - r i n g .
W e note the following result, the proof of which is omitted:
P R O P O S I T I O N 4.8. Let A be an/D-ring. Then A ' , under the strong order, is a lattice- ordered ring if and only if (1, O) A (0, a) exists/or every a e A + .
C H A P T E R I V . 0 N E - S I D E D / - I D E A L S I N f - R I N C S
I n C h a p t e r I I , a question was raised concerning the existence i n / - r i n g s of one-sided l-ideals t h a t are n o t two-sided. A t t h a t time, we p r e s e n t e d a n e x a m p l e of a n ordered ring containing a r i g h t / - i d e a l t h a t is not two-sided. I n this chapter, we p r e s e n t a n e x a m p l e of a n ordered ring A w i t h o u t non-zero divisors of zero w i t h this p r o p e r t y , a n d we show t h a t if an ordered ring w i t h o u t non-zero divisors of zero has a right 1-ideal t h a t is not two-sided, t h e n it contains a subring isomorphic to A. I n Section 2, we consider the s t a t u s of this question for the larger classes o f / - r i n g s t h a t h a v e been considered in this paper.
208 D . G . JOHNSON 1. All example
E x a m p l e 1.1. L e t S d e n o t e t h e free s e m i g r o u p w i t h o u t i d e n t i t y g e n e r a t e d b y t w o e l e m e n t s a, x: t h e e l e m e n t s of S a r e " w o r d s " of t h e f o r m
z = a n l x G a n ' ~ x G . . . a n ~ x ~r,
(1)
w h e r e t h e n~ a n d k s are n o n - n e g a t i v e integers, a n d n 1 a n d ]~1 a r e n o t b o t h zero. W i t h e a c h e l e m e n t z of S, w h e r e z h a s t h e f o r m (1), a s s o c i a t e t h e n o n - n e g a t i v e i n t e g e r N ( z ) = ~ n~.
W e define a t o t a l o r d e r in S as follows. I f z E S is as in (1), a n d if z' = an'x~'an~xk~...
a x E ~, t h e n we will s a y t h a t z > z' if a n d o n l y if i) N ( z ) < N ( z ' ) , or
if) N ( z ) = N ( z ' ) a n d e i t h e r
1) n I = n;, k 1 = k;, n 2 = n g , k 2 = kg . . . km 1 = k/n-!, a n d n m < nm for s o m e m = 1, 2 . . . r - l , or
2) n l = n ~ , /c l= k ~ , n 2 = n ~ , / Q = k ~ . . . /c~ 1 = ] c ~ i, n m = n ~ , a n d k m>/c;n for some m = 1, 2, ..., r.
Thus, if z ~ z' a n d if N (z) = N (z'), t h e n t h e o r d e r r e l a t i o n b e t w e e n z a n d z' is d e t e r m i n e d b y o b s e r v i n g t h e first e x p o n e n t a t w h i c h t h e e x p r e s s i o n s of z a n d z' in t h e f o r m (1) differ.
I t is e a s i l y seen t h a t t h i s is a t o t a l o r d e r on S. Moreover, t h i s o r d e r is p r e s e r v e d u n d e r m u l t i p l i c a t i o n ; t h a t is, if z, z', z " E S , w i t h z > z', t h e n z z " > z ' z " a n d z " z > z " z ' . F o r , if N ( z ) < N ( z ' ) , t h e n N ( z z " ) = N ( z " z ) < N ( z " z ' ) = N ( z ' z " ) , a n d if N ( z ) = N ( z ' ) , t h e n N ( z z " ) = N ( z ' z " ) = N ( z " z ) = N ( z " z ' ) a n d w h i c h e v e r of t h e c o n d i t i o n s 1) a n d 2) of if) a b o v e holds for z, z' will also h o l d for z z " , z ' z " a n d z " z , z " z ' .
N o w let A d e n o t e t h e s e m i g r o u p r i n g of S o v e r t h e ring of integers. T h e e l e m e n t s of A are finite f o r m a l sums:
I n A , a d d i t i o n is d e f i n e d e o o r d i n a t e w i s e , a n d m u l t i p l i c a t i o n is d e f i n e d b y :
y k z i z ) = z k
I n o r d e r t o d e f i n e a n o r d e r in A , we w r i t e t h e e l e m e n t s of A in t h e f o r m ~ m~z~, where
i = 1
i < ?" implies z~ > zj. Now, s a y t h a t ~ m~z~ >~ 0 if a n d o n l y if m 1 > 0, or m I = 0 a n d m2 > 0,
i = 1
A S T R U C T U R E T H E O R Y F O R A C L A S S O F L A T T I C E - O R D E R E D R I N G S 209 or . . . , or m I - m2 . . . ran_ 1 = 0 a n d m~ >~ 0. I t is r e a d i l y v e r i f i e d t h a t t h i s is a t o t a l o r d e r in A , u n d e r w h i c h A is a n o r d e r e d ring. Clearly, A c o n t a i n s no non-zero divisors of zero.
I n A , <a>,. is a r i g h t /-ideal t h a t is n o t t w o - s i d e d , since (a>r =
{ d E A : ]d] ~< moa + a ~ m~z~, m o a n integer, ~ m~z~EA } a n d , clearly, xa~.(a>r.
i =1 i--1
T h e r e s t of t h i s section is d e v o t e d to t h e p r o o f of t h e following result.
T H E O R E ~ 1.2. I] A is an ordered ring without non-zero divisors o/zero that contains a right 1-ideal that is not two-sided, then A contains a subring that is isomorphic to the ring o/
E x a m p l e 1.1.
B y T h e o r e m I I I . 3 . 5 , t h e s t r o n g o r d e r m a k e s t h e r i n g e x t e n s i o n A 1 of A i n t o a n o r d e r e d r i n g w i t h o u t n o n - z e r o divisors of zero in w h i c h t h e o r d e r i n g e x t e n d s t h a t on A . l~ecalling t h e d e t a i l s of t h i s i m b e d d i n g , we n o t e t h a t if I is a n y r i g h t / - i d e a l in A , t h e n i t is a r i g h t (ring) i d e a l in A D a n d hence t h a t if I is p r o p e r in A , t h e n ( I } r is a p r o p e r r i g h t L i d e a l in A 1.
Now, if I is a r i g h t 1-ideal in A t h a t is n o t t w o - s i d e d , t h e n t h e r e are p o s i t i v e e l e m e n t s a E 1 a n d x C A such t h a t x a r I . Then, in A, (a}r is a r i g h t / - i d e a l t h a t is n o t t w o - s i d e d . H e n c e , as n o t e d a b o v e , t h e r i g h t 1-ideal J = ( a } r in A 1 is a r i g h t L i d e a l of A 1 t h a t is n o t t w o - s i d e d . Since A 1 c o n t a i n s a n i d e n t i t y element, J is a m o d u l a r r i g h t l-ideal. H e n c e , J can be e x t e n d e d t o a m a x i m a l ( m o d u l a r ) r i g h t / - i d e a l M of A 1 ( P r o p o s i t i o n II.1.2). B y C o r o l l a r y I I . 4 . 7 , M is a t w o - s i d e d / - i d e a l in A 1. Thus, J * - ( a } is a p r o p e r / - i d e a l in A1, since J * ___ M ; i t is t h e s m a l l e s t (two-sided) l-ideal of A 1 t h a t c o n t a i n s J .
N o w let J , d e n o t e t h e l a r g e s t l-ideal of A 1 t h a t is c o n t a i n e d in J . Since t h e r i g h t 1- ideals of A 1 form a chain, J * is t h e s m a l l e s t / - i d e a l of A 1 t h a t p r o p e r l y c o n t a i n s J , . H e n c e , A 1 / J , is a s u b d i r e c t l y i r r e d u c i b l e ]-ring, w i t h s m a l l e s t n o n - z e r o 1-ideal J * / J , . B y T h e o r e m I I . 5 . 2 , J * / J , consists e n t i r e l y of n i l p o t e n t e l e m e n t s of o r d e r two; t h a t is, z E J * i m p l i e s z ~ E J , .
Now, let I* = J * (1 A a n d I , - J , N A . W e h a v e : I , c ( a ) r ~- I ~ I*. Thus, we h a v e p r o v e d :
L E M ~ A 1.3. I / A is an ordered ring without non-zero divisors o/ zero that contains a right 1-ideal I that is not two-sided, then A is not l-simple. More precisely, there is an l-ideal
I , ~_ I and an 1.ideal I* ~_ I, with 1.2 ~ I , .
F o r t h e sake of e c o n o m y , we will write, for b, c E A 1, b ~ c in case b ~ n c for e v e r y i n t e g e r n. N o t e t h a t , since xa(~J, we h a v e x a ~ n a for e v e r y i n t e g e r n, whence x > ~ l in A 1.
T h e p r o o f of T h e o r e m 1.2 will p r o c e e d as follows. W e consider t h e s u b - s e m i g r o u p T of t h e m u l t i p l i c a t i v e s e m i g r o u p of A g e n e r a t e d b y a, x, a n d s h o w t h a t t h e o r d e r i n d u c e d in T b y t h a t in A is t h e s a m e as t h e o r d e r d e f i n e d on t h e s e m i g r o u p S of E x a m p l e 1.1. More-
210 D . G . JOHNSON
over, we will show t h a t if z, z' E T a n d z > z', t h e n z > ~ z ' . T h e n t h e s u b r i n g B of A g e n e r a t e d b y T is j u s t t h e s e m i g r o u p r i n g of T o v e r t h e r i n g of i n t e g e r s a n d t h e o r d e r on B is t h e s a m e as t h e o r d e r on t h e r i n g A of E x a m p l e 1.1. F o r , if ~ m ~ z ~ E B , w h e r e e a c h z~E T, e a c h m~
t = l
s a n integer, a n d i < ] i m p l i e s zf > z j , t h e n ~ m i z ~ > 7 0 if a n d o n l y if m 1 > 0 , or m 1 = 0
4=1
a n d m 2 > 0 , or ..., or m 1 = m 2 . . . r a n _ 1 = 0 a n d mn ~>0.
I n w h a t follows, we will m a k e t a c i t use of t h e following facts:
1) T h e e l e m e n t s of T are n o n - z e r o p o s i t i v e e l e m e n t s of A .
2) A c o n t a i n s no n o n - z e r o divisors of zero, hence T is c a n c e l l a t i v e . Thus, if z, z', z" E T a n d z z ' > z z " , t h e n z' > z " .
n. k, n~ . . anrxk~ Z' = a n : x k : a n: " "
L e t z, z ' E T , s a y z = a x a . a n d . . . a n s x ks. As before, l e t N ( z ) = ~ n~ a n d N ( z ' ) = ~ n~. T h a t N ( z ) is u n i q u e l y d e t e r m i n e d for each z e T is shown
i = l ] = 1
b y I) below.
X k~ Z p = X k:.
I f N (z) = N (z') = 0, t h e n z = a n d Since, in A1, x ~ 1, we h a v e z > z' if a n d o n l y if k 1 > k~, in w h i c h case z > ~ z ' . I f N ( z ) = 0 a n d N ( z ' ) 4 0, t h e n , in A1, z' e J*, w h e n c e
z ' ~ I ~ z = x k'. Thus, if e i t h e r N ( z ) or N ( z ' ) is zero, t h e n t h e o r d e r r e l a t i o n b e t w e e n z a n d
z ' is t h e s a m e as t h a t d e f i n e d in E x a m p l e 1.1. I n w h a t follows, we will consider t h o s e zE T w i t h N (z) >/1.
I) I f N ( z ) < N ( z ' ) , t h e n z > ~ z ' .
W e p r o v e t h i s b y i n d u c t i o n on N(z). I f N ( z ) = 1 a n d N ( z ' ) > 1, t h e n z >~a, so z ~ i I . ,
a n d z' m a y be w r i t t e n as t h e p r o d u c t of t w o e l e m e n t s of I * , whence z ' E I , . Since I , is a n /-ideal in A , z > ~ z ' .
N o w s u p p o s e t h a t t > 1 a n d t h a t I) is t r u e w h e n e v e r iV(z) < t. I f N ( z ) = t a n d N ( z ' ) > t,
t h e n we consider t w o cases:
i) k.>~ k;. F o r a n y i n t e g e r n,
nr X k r - ks - - " " " " "
z - - n u ' = ( a n` x k' . . . a n a n ~ x k' . . . a ns) x %
~ (an~ Xk~ . . . a n t _ n a n ~ x k ~ . . . a n s ) x k s
. . . - - n a n ' x k~ . . . a ns 1) a x ks >~ 0 (an~ Xk~ a n t - 1
b y t h e i n d u c t i o n h y p o t h e s i s . ii) k~< ]c~. F o r a n y i n t e g e r n,
z - n z ' = ( a n` x k~ . . . a n r - - It a n. x k' . . . a ns x ks - k~.) x k , .
A S T R U C T U R E T H E O R Y :FOR A CLASS OF L A T T I C E - O R D E R E D RI~TGS 2 1 1
N o w , since x a > a x , w e h a v e
" X k : " " - " X k : a n s - 1 x k S - kr
an~ ... ans xkS kr ~ an~ . . . a,
so z -- n z' > ( a nl x kl ... a n r - 1 _ n a n: x k: . . . a n'~ 1 xk~ - k,) a x kr >~ 0 b y t h e i n d u c t i o n h y p o t h e s i s .
T h u s , i n e a c h case, z >~ z ' , so I) is e s t a b l i s h e d .
I I ) I f N ( z ) = N ( z ' ) , n l = n ; , ]Cl = It;, n 2 = n ~ .. . . , ]Cm_l=km_l, a n d n m < n ~ n , t h e n z > z ' . T h e p r o o f is a g a i n b y i n d u c t i o n o n i V ( z ) . I f i V ( z ) - l , t h e n t h e r e is o n l y o n e case: z = x kl a x k~ a n d z ' = a x k:, w h e r e / c I ~ 0. T h e n z >~ x a (~ I , w h i l e z ' E I , w h e n c e z > z'.
N o w s u p p o s e t h a t t > l , a n d t h a t I I ) h o l d s w h e n e v e r i V ( z ) < t . L e t z, z' E S s a t i s f y t h e h y p o t h e s i s of I I ) , w i t h N ( z ) = t . T h e r e a r e f o u r cases:
i) m ~ 1. F o r a n y i n t e g e r n,
z - n z ' = a n ' x k l a ( a n ~ - l x k~ . . . x k ' - n a ~ ; - l x k~ . . . xkS)>~ 0 b y t h e i n d u c t i o n h y p o t h e s i s .
ii) m = 1 a n d n 1 4 0 . F o r a n y i n t e g e r n,
z - n z' = a (a n' - 1 xk~ ... xk, _ n a n; - 1 xk: ... x~:) 1> 0 b y t h e i n d u c t i o n h y p o t h e s i s .
iii) r e = l , n 1 = 0 , a n d n ~ 4 1 . F o r a n y i n t e g e r n,
z - n z ' = x kl a n~ x k~ . . . x k" - n a n: x k: . . . x ks.
N o w , s i n c e x a ~ a x , w e h a v e
x k' a n~ . . . x ~r > a x k, a n~ - 1 . . . Xkr,
w h e n c e z -- n z' > a (x ~' a n~- 1 ... x k, _ n a n:- 1 xk: ... xk~) >~ 0 b y t h e i n d u c t i o n h y p o t h e s i s , s i n c e n l - 1 > 0.
i v ) m = 1, n l = O , a n d n l = l . A s in t h e p r o o f of I), w e c o n s i d e r t w o cases:
1) /c~>k:. F o r a n y i n t e g e r n,
z - - n z ' = ( x k ' a n~ x k~ . . . a n" x kr k s - - n a x k" a n~ . . . a n s ) x k s
(xklan~ x k" ... a n ' - - n a x k : a n: . . . an'S) xkS
= ( x k ' a n ~ x k~ . . . a n ' - I _ n a x ~ : a n : . . . a n s - 1 ) a x k ' s > ~ O b y t h e i n d u c t i o n h y p o t h e s i s .
1 4 - 60173033. A c t a m a t h e m a t i c a . 104. I m p r i m 6 le 20 d 6 c e m b r e 1960
212 D. G. JOIt~SO~
2) ]Cr< k:. F o r a n y i n t e g e r n,
. . . . . . - kr) X k r
Z - - ZtZ' : ( x k l a n~ a " r - n a x k ~ a ~ a n s x ks
( x kl a n~ . . . a nr - - Tb a x kl a n2 . . . a ns - 1 x k S - kr a ) X kr
n , - 1 " " " - l x k i - k r )
= ( x k l a n2 . . . a - - n a x k ~ a n2 . . . a ns a x ~ ' ~ O
b y t h e i n d u c t i o n h y p o t h e s i s .
T h u s , i n e a c h case, we h a v e z >~ z', so ] I ) is e s t a b l i s h e d .
l p 9 p
I I I ) I f N ( z ) = N ( z ' ) , n 1 = n l , IQ = k l , n ~ = n2 . . . n,~ = n m , a n d / c m > k~, t h e n z >~ z . T h e p r o o f is a g a i n b y i n d u c t i o n o n N ( z ) . F o r N ( z ) = 1, t h e r e a r e t w o cases;
k 2 Z p
i) z = x k' a x , x ~ a x k', w i t h k 2 > k~. F o r a n y i n t e g e r n , z - n z ' = x k' a x k' ( x k ~ - k, _ n ) > O.
ii) z = x a x , x a x , w i t h k l > / O . F o r a n y i n t e g e r n ,
x ~: ~ " a x k' ~ I , w h i l e a x k~ e I .
z - n z ' ~ ( x ~ k ~ a x k ~ - - n a x ~ ' ) > O , since x k~-k:
N o w s u p p o s e t h a t t > 1 a n d I I I ) h o l d s w h e n e v e r N ( z ) < t . L e t z , z ' E S s a t i s f y t h e h y p o t h e s i s of I I I ) , w i t h N ( z ) = t. T h e r e a r e t w o eases:
i) n l = # 0 . F o r a n y i n t e g e r n,
z _ n z , = a ( a n , - l x k l a n ~ . .x k ~ _ n a n , - l x . . . . k[ X 1~s) >~0
b y t h e i n d u c t i o n h y p o t h e s i s .
ii) n l = 0 . W e c o n s i d e r t w o cases:
1) k 1 =/c~, F o r a n y i n t e g e r n,
z - n z' = x k' a (a ~ - 1 xk, ... xk~ _ n a ~ - 1 x~ ... xk~) > / 0 b y t h e i n d u c t i o n h y p o t h e s i s .
2) /c1>/c~. F o r a n y i n t e g e r n ,
. . . n a ~ x k~ ... x ~)~>0 b y I I ) . z - n z ' = x kl ( x ~ ' - k, an~ x k , _ k"
T h u s , m e a c h case, z >~ z' so I I I ) is e s t a b l i s h e d .
I), I I ) , a n d I I I ) s h o w t h a t t h e o r d e r i n T is e x a c t l y t h e o r d e r g i v e n for t h e s e m i g r o u p S i n E x a m p l e 1.1. A s n o t e d e a r l i e r , t h i s c o m p l e t e s t h e p r o o f of T h e o r e m 1.2.
A S T R U C T U R E T H E O R Y - F O R A G L A S S O F L A T T I C E - O R D E R E D RIbYGS 213
2. The general question
T h e result stated in L e m m a 1.3 can be s t a t e d in t h e following more general form:
P R 0 P 0 S I T I 0 ~ 2 . 1 . Every ordered ring that contains a non-zero proper one-sided 1-ideal
contains a non-zero proper two-sided l-ideal.
Proo]. F o r ordered rings w i t h o u t non-zero divisors of zero, this follows f r o m L e m m a 1.3 a n d its left analogue. I f A is an ordered ring containing non-zero divisors of zero b u t no non-zero proper 1-ideals, t h e n A contains non-zero nilpotents b y Theorem 1.4.1, hence Z~(A) = ( a E A : a 2= O} = A , since Z2(A ) is an /-ideal in A (cf. t h e discussion following Corollary 1.3.17). Then, since e v e r y / - s u b g r o u p of t h e additive group of A is an l-ideal in A, there are no non-zero p r o p e r / - s u b g r o u p s of the additive g r o u p of A. Hence, there are no non-zero proper one-sided/-ideals in A.
As a n immediate consequence, we have:
COROLLARY 2.2. Every maximal l-ideal in an ]-ring A is a maximal right (left) 1-ideal
o/A.
I n the remainder of this section, we summarize w h a t is k n o w n a b o u t the existence of one-sided 1-ideals t h a t are n o t two-sided in t h e general classes of ]-rings t h a t have been considered in this paper.
Recall t h a t the ring A of E x a m p l e II.6.2 is an ordered ring containing only one non- zero p r o p e r two-sided 1-ideal (Fz). Hence, A satisfies the descending chain condition for 1-ideals, a n d A is subdirectly irreducible, so we have:
l) There exist ]-rings satisfying t h e descending chain condition for l-ideals t h a t contain one-sided 1-ideals t h a t are n o t two-sided.
2) There exist subdirectly irreducible ]-rings t h a t contain one-sided 1-ideals t h a t are n o t two-sided.
Since the ring of E x a m p l e 1.1 contains no non-zero divisors of zero, we have:
3) There exist ]-rings with zero 1-radical t h a t contain one-sided 1-ideals t h a t are n o t two-sided.
I f A is an ]-ring with z e r o / - r a d i c a l t h a t satisfies the descending chain condition for 1-ideals, then, b y L e m m a II.5.7, ~t is isomorphic to a (finite) direct union of/-simple ordered rings A1, ..., An. B y Proposition 2.1, each M~ contains no non-zero proper one-sided 1- ideals. If I is a right (left) 1-ideal in A, t h e n I f3 M i is a right (left) l-ideal in A~ for i = 1, .... n. Hence, each I f3 A~ is {0) or A~, so I is clearly a two-sided 1-ideal in .4. Thus, we have:
4) I f A is an ]-ring with z e r o / - r a d i c a l which satisfies t h e descending chain condition for/-ideals, t h e n every one-sided/-ideal of A is two-sided.
214 D.G. JOHI~SON
Since t h e J - r a d i c a l of a n / - r i n g a l w a y s c o n t a i n s t h e / - r a d i c a l , we have:
5) I f A is a J - s e m i s i m p l e /-ring w h i c h satisfies t h e d e s c e n d i n g c h a i n c o n d i t i o n for /-ideals, t h e n e v e r y o n e - s i d e d / - i d e a l of A is t w o - s i d e d .
Our final e x a m p l e , w h i c h is d u e t o P r o f e s s o r H e n r i k s e n , shows t h e following:
6) T h e r e e x i s t J - s e m i s i m p l e /-rings t h a t c o n t a i n o n e - s i d e d / - i d e a l s t h a t a r e n o t t w o - sided.
E x a m p l e 2.3. L e t A be a n o r d e r e d r i n g w i t h i d e n t i t y a n d w i t h o u t n o n - z e r o d i v i s o r s of zero t h a t c o n t a i n s a r i g h t / - i d e a l I t h a t is n o t t w o - s i d e d (cf. S e c t i o n 1). L e t A [2] d e n o t e t h e r i n g of p o l y n o m i a l s in one i n d e t e r m i n a t e 2 w i t h coefficients in A , o r d e r e d l e x i c o g r a - p h i c a l l y w i t h t h e l e a d i n g coefficient d o m i n a t i n g : a 0 + a12 + 9 9 9 + an2 n >~ 0 if a n d o n l y if a n > 0 , or a n = 0 a n d an_ 1 > 0 , or ..., or a n = a n _ l . . . a 1 = 0 a n d a 0~>0. W i t h t h i s o r d e r i n g , A [2] is a n o r d e r e d r i n g w i t h i d e n t i t y a n d w i t h o u t n o n - z e r o divisors of zero;
i t is c l e a r l y / - s i m p l e , since l a21 > 1 for e v e r y 0 =~ a EA. H e n c e , A [2] is a n / - p r i m i t i v e / - r i n g . O b s e r v e t h a t A is a n o r d e r e d s u b - r i n g of A[2] a n d t h a t , if y E A [ 2 / , t h e n y ~ A i m p l i e s
[y [ > [ a [ for e v e r y a e A .
N o w l e t B d e n o t e t h e set of all sequences in A [2] t h a t a r e e v e n t u a l l y in A : if z E B , t h e n z = (zl, z 2 .. . . ), where e a c h zi EA [2], a n d t h e r e is a n i n t e g e r n such t h a t i ~> n i m p l i e s z~ EA. U n d e r c o o r d i n a t e w i s e r i n g a n d l a t t i c e o p e r a t i o n s , B is a n / - r i n g ; i t is c l e a r l y a sub- d i r e c t u n i o n of copies of t h e / - p r i m i t i v e / - r i n g A [2]. H e n c e , B is a J - s e m i s i m p l e / - r i n g w i t h i d e n t i t y .
L e t J = ( z E B : z is e v e n t u a l l y in I } . Since I is a r i g h t / - i d e a l in A t h a t is n o t t w o - sided, i t is clear t h a t J is a r i g h t / - i d e a l in B t h a t is n o t t w o - s i d e d .
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Received June 11, 1959