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The quotient ring of the ring of a mortiacontextF.A.M. Al-Tayar aa School of Mathematics , University of Leeds , Leeds, LS2 9JT, EnglandPublished online: 27 Jun 2007.
To cite this article: F.A.M. Al-Tayar (1982) The quotient ring of the ring of a mortia context, Communications inAlgebra, 10:6, 637-654, DOI: 10.1080/00927878208822740
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COMMUNICATIONS IN ALGEBRA, 10 (6), 637-654 (1982)
THE QUOTIENT R I N G OF THE R I N G OF A MORTIA
CONTEXT
F.A.M. A l - T a y a r
S c h o o l o f M a t h e m a t i c s , U n i v e r s i t y o f L e e d s L e e d s LS2 9JT. E n g l a n d .
I n t h i s p a p e r , we g i v e n e c e s s a r y and s u f f i c i e n t
R V c o n d i t i o n s f o r t h e r i n g T = ( S ) , o f a ~ o r i t a
W
C o n t e x t , t o h a v e a r i g h t A r t i n i a n r i g h t ( c l a s s i c a l )
q u o t i e n t r i n g , a n d t o h a v e a t w o - s i d e d N o e t h e r i a n
o n e , p r o v i d e d i n t h e l a t t e r c a s e , T i s N f i e t h e r i a n
( l e f t a n d r i g h t ) a n d t h e b i m o d u l e s RVS a n d S W R a r e
t o r s i o n - f r e e . We b e g i n i n S e c t i o n (I), b y r e c a l l i n g
p r e l i m i n a r y r e s u l t s c o n c e r n i n g A r t i n i a n a n d N o e t h e r i a n
q u o t i e n t r i n g s . S e c t i o n ( 2 ) i s d e v o t e d t o t h e
d e f i n i t i o n a n d some p r o p e r t i e s o f t h e r i n g o f a M o r i t a
C o n t e x t . S e c t i o n ( 3 ) c o n t a i n s t h e ma in two t h e o r e m s
( 3 . 1 , a n d 3 .4 ) o f t h i s P a p e r .
Copyright O 1982 by Marcel Dekker. Inc.
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N o t a t i o n and C o n v e n t i o n . A l l r i n g s w i l l h a v e u n i t
e l e m e n t , a l l m o d u l e s a n d b i m o d u l e s a r e u n i t i a l ,
" Q u o t i e n t r i n g " , w i l l mean q u o t i e n t r i n g w i t h r e s p e c t
t o t h e s e t o f a l l r e g u l a r e l e m e n t s . J (R) , and N ( R )
w i l l d e n o t e t h e J a c o b s o n and n i l p o t e n t y a d i c a l s ,
r e s p e c t i v e l y , o f t h e r i n g R. I f V i s a r i g h t ( l e f t )
R-module, K a s u b s e t o f R , and F a s u b s e t o f V , t h e n
L(VR) rill d e n o t e t h e l a t t i c e o f a l l R-submodules o f
v ~ ' r R ( K ) ( L R ( K ) ) w i l l d e n o t e t h e r i g h t ( l e f t )
a n n i h i l a t o r o f K i n R , r ( F ) ( L R ( F ) ) w i l l d e n o t e t h e R
r i g h t ( l e f t ) a n n i h i l a t o r o f F i n R , a n d L v ( K ) ( r V ( k ) )
w i l l d e n o t e t h e l e f t ( r i g h t ) a n n i h i l a t o r o f K i n V .
F i n a l l y , i f I i s a r i g h t i d e a l o f R , t h e n C R ( I ) w i l l
d e n o t e t h e s e t o f a l l e l e m e n t s o f R r e g u l a r modulo I .
SECTION (1)
L e t R b e a r i n g , and M b e a r i g h t R-mcdule . Then M R
i s s a i d t o b e t o r s i o n - f r e e ( w i t h r e s p e c t t o C R ( 0 ) ) ,
i f mc # 0 f o r e v e r y 0 # rn E M a n d e v e r y c i n C R ( 0 ) .
S u p p o s e R h a s a r i g h t q u o t i e n t r i n g Q , a n d l e t M b e
a t o r s i o n - f r e e R-module. Then M h a s a q u o t i e n t Q -
module M Q . M Q i s c o n s t r u c t e d f rom M i n a s i m i l a r
f a s h i o n a s Q i s c o n s t r u c t e d f rom R . The e l e m e n t s
o f M Q h a v e t h e form mc'l w i t h m E M a n d c E C ( 0 ) . R
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Lemma 1.1 ( [ I ] , P r o p o s i t i o n 1 . 5 ) . L e t R b e a r i n g
w i t h a r i g h t q u o t i e n t r i n g Q . L e t M b e a n R-module
which i s t o r s i o w f r e e . Then M Q = M 0 Q , u n d e r t h e R
c o r r e s p o n d e n c e mq + m q .
Lemma 1 . 2 ( [ ? I , ( 1 . 3 6 ) , p . 2 3 4 ) . L e t R be a r i n g w i t h
a r i g h t q u o t i e n t r i n g Q. I f cl,. .., c a r e r e g u l a r k
e l e m e n t s o f R , t h e n t h e r e e x i s t s r e g u l a r e l e m e n t s ,
- 1 - 1 c , d l , . . , d k o f R s u c h t h a t c i = ddjc , f o r i=l ,...,k.
Theorem 1 . 3 ( [ 4 1 , Theorem 2 . 1 0 ) . L e t R b e a r i n g .
Then R h a s a r i g h t A r t i n i a n r i g h t q u o t i e n t r i n g i f
and o n l y i f R s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n s :
( i ) ' R and R / N a r e r i g h t G o l d i e r i n g s , w h e r e
N = N(R).
k (ii) R / L ( N ) i s a r i g h t f i n i t e - d i m e n s i o n a l r i n g
R
f o r k = 1, ..., p - 1 whe re p i s a n i n t e g e r
s u c h t h a t N~ = 0 b u t N ~ ' ~ z 0 .
Theorem 1 . 4 ( [=I , Theorem 3) . L e t R b e a r i n g w i t h
a r i g h t A r t i n i a n r i g h t q u o t i e n t r i n g Q . L e t e b e
a n o n - z e r o i d e m p o t e n t e l e m e n t o f R. Then eRe h a s a
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r i g h t A r t i n i a n r i g h t q u o t i e n t r i n g wh ich i s i s o m o r p h i c
t o e Q e .
Lemma 1 . 5 ( [ ? I , Theorem 1) . I n a r i g h t ( l e f t ) G o l d i e
r i n g , e v e r y n i l s u b r i n g i b n i l p o t e n t .
L e t R b e any r i n g . We d e f i n e t h e A r t i n i a n r a d i c a l A ( R )
o f R t o b e t h e sum o f a l l t h e A r t i n i a n r i g h t i d e a l s o f
R ( b y a n A r t i n a n r i g h t i d e a l we mean a r i g h t i d e a l
which i s A r t i n i a n a s a r i g h t R-module). We h a v e t h e
f o l l o w i n g .
Theorem 1 . 6 ( [ G I , Theorem 2 . 6 ) . L e t R b e a
N ~ e t h e r i a n r i n g . Then R i s e q u a l t o i t s own f u l l
( o r two s i d e d ) q u o t i e n t r i n g i f a n o n l y i f
C o r o l l a r y 1 .7 ( [UI. C o r o l l a r y 2 . 7 ) . L e t R b e
a N o e t h e r i a n r i n g t h a t i s e q u a l t o i t s own f u l l
q u o t i e n t r i n g . Then R is s e m i - l o c a l .
F i n a l l y , by a s t r a i g h t f o r w a r d m o d i f i c a t i o n t o t h e
p r o o f o f t h e ma in r e s u l t i n [ 6 1 , o n e c a n g e t t h e - f o l l o w i n g .
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Lemma 1 . 8 . L e t RMS b e a b i m o d u l e s u c h t h a t RM h a s
f i n i t e l e n g t h , a n d M i s N o e t h e r i a n . Then MS h a s S
f i n i t e l e n g t h .
SECTION ( 2 )
L e t R,S b e r i n g s , W b i m o d u l e s w i t h R'S' S R
homomorphisms V Q W + R , W 8 V + S ( c a l l e d S R
m u l t i p l i c a t i o n ) , s u c h t h a t T = ( R V i s a r i n g .
W S
T i s c a l l e d t h e r i n g o f a M o r i t a C o n t e x t , S e e [L].
e . g . I f R i s a r i n g , n a n y n a t u r a l number, t h e n t h e
r i n g s M, ( R ) a n d Tn ( R ) , o f n x n m a t r i c e s a n d o f n x n
u p p e r ( l o w e r ) t r i a n g u l a r m a t r i c e s o v e r R ,
r e s p e c t i v e l y , c a n b e r e g a r d e d a s t h e a p p r o p r i a t e
r i n ' g s o f M o r i t a C o n t e x t s .
P r o p o s i t i o n 2 . 1 . TT i s N o e t h e r i a n ( A r t i n i a n ) i f
and o n l y i f R R , S S P VS a n d W a r e N o e t h e r i a n R
( A r t i n i a n ) .
P r o o f . I f I1 I2 .. . , a n d V1 t V2 .. ., a r e
r e s p e c t i v e l y a s c e n d i n g c h a i n s o f r i g h t i d e a l s o f R
and S - submodu le s o f V , t h e n e l l I I T c e l l 1 2 T s.. . , and e l l V I T s e l l V 2 T .. . , a r e a s c e n d i n g c h a i n s o f
r i g h t i d e a l s o f T, whe re eH T h e r e f o r e ,
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i f TT i s N o e t h e r i a n , t h e n R R and VS a r e a l s o
N o e t h e r i a n . S i m i l a r a rguments w i l l show t h a t SS
and W R a r e N o e t h e r i a n . For t h e c o n v e r s e , i f we
s e t A = ( R V 0 0
1 , B = ( W S ) l and C = R
O ) I t h e n 0 0 ( 0 S
A and B a r e r i g h t i d e a l s o f T and A and B a r e C-
modules. I n f a c t AC and B a r e f . g . . C
Thus A C
and B a r e N o e t h e r i a n . S i n c e L(A c L ( A C ) , and C T -
L ( B T ) g. L ( B C ) . Then AT and B a r e N o e t h e r i a n . T
T h e r e f o r e TT i s N o e t h e r i a n . The A r t i n i a n c a s e can
be p roved i n a s i m i l a r way.
Next we need t h e f o l l o w i n g Lemma.
J ( R ) M Lemma 2 . 2 . ( [ 8 ] , Theorem 8 ) . J ( T ) = (
J ( S ) ) ' where
I f RMS i s a b imodule . Then M i s s a i d t o s a t i s f y t h e
r i g h t ( l e f t ) Ore c o n d i t i o n ( w i t h r e s p e c t t o C R ( 0 )
and C S ( 0 ) ) , i f f o r e v e r y c E C R ( 0 ) ( d E CS ( 0 ) and
e v e r y m EM(^' E M ) , t h e r e e x i s t d E CS(0) ( C E C ( 0 ) ) R
and m ' E M ( m E M ) such t h a t
cm' = md.
RMS i s t o r s i o n - f r e e , w i l l mean M i s t o r s i o n - f r e e R
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RING OF A MORTIA CONTEXT 64 3
w i t h r e s p e c t t o C R ( 0 ) , and M i s t o r s i o n - f r e e w i t h S
r e s p e c t t o C ( 0 ) . An i m m e d i a t e c o n s e q u e n c e o f t h e S
a b o v e d e f i n i t i o n s and Lemma 1.1 i s t h e f o l l o w i n g .
R V Lemma 2 . 3 . L e t T = ( W S ) , b e t h e r i n g o f a M o r i t a
C o n t e x t . I f
( i ) R and S h a v e r i g h t q u o t i e n t r i n g s Q ( R ) and
Q ( S ) r e s p e c t i v e l y ,
( ii) RVS and S W R a r e t o r s i o n - f r e e , a n d
(iii) RVS and S W R s a t i s f y t h e r i g h t Ore c o n d i t i o n
w i t h r e s p e c t t o C R ( 0 ) and CS ( 0 )
t h e n t h e s e t
forms a r i n g .
To end t h i s s e c t i o n , two lemmas a r e r e q u i r e d .
R V Lemma 2 . 4 . L e t T = (W S ) , b e t h e r i n g o f a M o r i t a
C o n t e x t . L e t T b e a s emi -p r ime r i g h t G o l d i e r i n g ,
a n d l e t RVS, S W R b e t o r s i o n - f r e e . S e t e = (i :).
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I f t i s a r e g u l a r e l e m e n t o f T, t h e n t h e r e
e x i s t s a n e l e m e n t b i n T s u c h t h a t e t b e i s
r e g u l a r i n R.
P r o o f . S i n c e RVS and - S W R a r e t o r s i o n - f r e e , t h e n an
e l e m e n t c c C R ( 0 ) i f f c + l - e = ( d !) p C T ( 0 ) . The re -
f o r e w e can u s e t h e same method a s i n p r o o f o f Lemma
4 . 4 . o f [GI t o g e t t h e above r e s u l t .
An i d e a l I o f a r i n g R i s s a i d t o b e ( r i g h t )
l o c a l i s a b l e i f C R ( I ) s a t i s f i e s t h e ( r i g h t ) Ore
c o n d i t i o n .
Lemma 2 . 5 . L e t T R V
( W S ) , b e t h e r i n g o f a M o r i t a
C o n t e x t s u c h t h a t b o t h .RVS and W a r e t o r s i o n - f r e e . S R
S e t e = (i ; I . I f I i s a s e m i p r i m e ( r i g h t )
l o c a l i s a b l e i d e a l o f T I t h e n e I e i s a s emi -p r ime
( r i g h t ) l o c a l i s a b l e i d e a l o f R .
P r o o f . Same p r o o f a s i n p r o p o s i t i o n 4 . 5 o f [GI .
SECTION ( 3 )
Our f i r s t r e s u l t i s t h e f o l l o w i n g .
Theorem 3.1. L e t T = ( W i) , b e t h e r i n g o f a
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M o r i t a C o n t e x t s u c h t h a t b o t h VS and W a r e f . g . . R
Then T i s a r i g h t o r d e r i n a r i g h t A r t i n i a n r i n g
Q ( T ) , s a y , i f and o n l y i f
( i) R and S a r e r i g h t o r d e r s i n r i g h t A r t i n i a n r i n g s ,
Q ( R ) and Q(S) r e s p e c t i v e l y
( i i) RVS and S W R a r e t o r s i o n - f r e e .
(iii) RVS and S W R s a t i s f y t h e r i g h t Ore c o n d i t i o n
w i t h r e s p e c t t o C ( 0 ) and CS ( 0 ) . . R
Moreover i f t h i s happens , t h e n
P r o o f . Assume we h a v e c o n d i t i o n s ( i ) , (ii) and (iii) . S e t H 4s .\a lemma 2 .3 above. Thus H forms a r i n g .
A l s o by o u r a s s u m p t i o n s t h a t V and W a r e f . g . , S R
c o n d i t i o n ( i) , and P r o p o s i t i o n 2 . 1 , we have H i s
r i g h t A r t i n i a n . Again by c o n d i t i o n s (ii) , ( i i i ) ,
Lemma 1.1, Lemma 1.2 and Lemma 2 . 3 , H i s a
c 0 l o c a l i z a t i o n o f T o v e r t h e se t L = { ( o d ) '
C E C R ( 0 ) , d & C S ( 0 ) ) . Now i f x E C (O) , t h e n x T
i s r i g h t r e g u l a r i n H. So x i s i n v e r t i b l e i n H ,
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ff
b e c a u s e H i s r i g h t A r t i n i a n . T h e r e f o r e H i s a
r i g h t q u o t i e n t r i n g o f T .
C o n v e r s e l y , assume t h a t Q (T) e x i s t s , and i s r i g h t A r t i n i a n
1 0 0 0 S e t O) , e ~ 2 = ( ~ l ) . Then e l l and e22 a r e non-
z e r o i d e m p o t e n t e l e m e n t s of T . Moreoever R = e l l T e l l ,
and S = e Z 2 T e z 2 . T h e r e f o r e Q ( R ) and Q ( S ) e x i s t
and t h e y a r e r i g h t A r t i n i a n , by Theorem 1 . 4 . S i n c e
R,S and T a r e r i g h t G o l d i e r i n g s by Theorem 1 . 3 , t h e n
N ( R ) ; N ( S ) , and N(T) a r e n i l p o t e n t i d e a l s o f R,S, and
T r e s p e c t i v e l y , by Lemma 1 .5 . So i t s now e a s y t o
show t h a t
Now i f , c E C ( 0 ) and d c C S ( 0 ) , t h e n c s C R ( N ( R ) ) R
c 0 and d E C S ( N ( S ) ) , by Theorem 1 . 3 . So ( O c C T ( N ( ~ ) )
= C T ( 0 ) . T h e r e f o r e i t i s c l e a r now t h a t RVS and W S R
a r e t o r s i o n - f r e e . To p r o v e c o n d i t i o n (iii) , l e t
Q 3 Q ( T ) , and I = T J ( Q ) . S i n c e Q i s s e m i l o c a l ,
then^ (0) = f u n i t s ) = C Q ( J ( Q ) . T h e r e f o r e ~ ~ ( 0 ) = Q
C T ( I ) . So I i s ( r i g h t ) l o c a l i s a b l e i n T , and by
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G o l d i e f s Theorem, T / I i s a semipr ime r i g h t G o l d i e
r i n g . S e t e = 0
( O 1 . Then d E C S ( 0 ) i f f d + l - e =
l O ) E CT(0) i f f d + l - e E C T ( I ) i f f d E CS ( e I e ) , by
(0 d
Lemma 2 . 4 and Lemma 2 . 5 . Now l e t c E C R ( 0 ) , v E V . We
want t o f i n d v f E V , d E CS ( 0 ) such t h a t c v ' = vd. A s
C E C T ( 0 ) . SO t h e r e e x i s t s f = b e f o r e ( O m ) ,
( n Y
h = t , i n T w i t h h € C T ( O ) s u c h t h a t ( W d
S i n c e h E C T ( 0 ) and C T ( 0 ) = C T ( I ) , t h e r e e x i s t s
b = b l l b12
by Lemma 2 . 4 . M u l t i p l y b o t h s i d e s of ( * ) by t h e
e l e m e n t b e , t o g e t t h e f o l l o w i n g
S e t v ' = x b l p + mb2p, d f = wblp+ dbpz E CS ( 0 ) . T h i s
g i v e s t h a t RVS s a t i s f i e s t h e r i g h t Ore c o n d i t i o n
w i t h r e s p e c t t o C R ( 0 ) and CS ( 0 ) . S i m i l a r a rguments
w i l l show t h a t S W R s a t i s f i e s t h e r i g h t Ore c o n d i t i o n
w i t h r e s p e c t t o CS(0) and C ( 0 ) . R
F i n a l l y i f we s e t H as i n t h e f i r s t p a r t of t h e p r o o f ,
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t h e n H i s a r i g h t A r t i n i a n r i g h t q u o t i e n t r i n g o f T.
T h e r e f o r e H 2 Q (T) .
C o r o l l a r y 3 . 2 ([9], Theorem 2 . 2 8 ) . L e t R b e a r i n g
which i s a r i g h t o r d e r i n a r i g h t A r t i n i a n r i n g Q ,
s a y . Then Mn(R), t h e r i n g o f n x n m a t r i c e s o v e r R.
i s a r i g h t o r d e r i n a r i g h t A r t i n i a n r i n g Q ( M n ( R ) .
Moreover Q ( M n ( R ) 1 = M n (Q) .
C o r o l l a r y 3 .3 . ( A c o n v e r s e t o Theorem 1 . 4 ) . L e t R b e
a r i n g w i t h a n o n - z e r o i d e m p o t e n t e . Then R i s a
r i g h t o r d e r i n a r i g h t A r t i n i a n r i n g i f and o n l y i f .
* ( i) t h e two r i n g s eRe and eRe a r e r i g h t o r d e r s i n
r i g h t A r t i n i a n r i n g s , 2 = 1-e .
* (ii) The b i m o d u l e s ( e ~ e ( eRe)* * and * *
e Re e R e eRe e Re
a r e t o r s i o n - f r e e and s a t i s f y t h e r i g h t Ore
c o n d i t i o n w i t h r e s p e c t t o C ( 0 ) and C;R;(0). e Re
P r o o f . We have
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Thus t h e r e s u l t becomes c l e a r .
The n e x t t h e o r e m d e s c r i b e when t h e r i n g o f a M o r i t a
C o n t e x t h a s a N o e t h e r i a n f u l l q u o t i e n t r i n g .
Theorem 3 . 4 . L e t T = ( W S ) , b e a ( l e f t a n d r i g h t )
N o e t h e r i a n r i n g s u c h t h a t b o t h RVS and W a r e t o r s i o n - s R
f r e e . Then T h a s a f u l l q u o t i e n t r i n g Q ( T ) , s a y , i f
and o n l y i f
( i ) R and S h a v e f u l l q u o t i e n t r i n g s Q ( R ) a n d Q(S)
r e s p e c t i v e l y , s a y
(ii) RVS and S W R s a t i s f y t h e l e f t and r i g h t Ore
c o n d i t i o n s w i t h r e s p e c t t o ' C R ( 0 ) a n d CS ( 0 ) .
Moreoever i f t h i s h a p p e n s , t h e n
P r o o f . Assume we h a v e c o n d i t i o n s ( i ) a n d ( i i) .
F i r s t we assume t h a t R = Q ( R ) a n d S = Q ( S ) . We
w a n t t o show t h a t T = Q ( T ) .
A AV O O ) , L e t A = A ( R ) , B = A ( s ) . S e t I ( O 0 ) ' =(BW B
O) = R @ 5 . S i n c e RV i s K = I $ J , a n d F = ( O
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N o e t h e r i a n , by P r o p o s i t i o n 2 . 1 , t h e n R ( A ~ ) i s
A r t i n i a n . So RI i s A r t i n i a n . Bu t IF i s N o e t h e r i a n .
~ h u s IF i s A r t i n i a n , by Lemma 1 . 8 . S i n c e L ( I T ) E
L ( I F ) , t h e n I i s A r t i n i a n . S i m i l a r a r g u m e n t s w i l l T
show t h a t J i s A r t i n i a n . T h e r e f o r e , KT i s A r t i n i a n . T
B u t K i s a r i g h t i d e a l o f T. Then K A ( T ) .
T h e r e f o r e
a T ( ~ ( T ) n r R ( A ( T ) aT(K) n r T ( K )
I t i s s t r a i g h t f o r w a r d b c h e c k t h a t
BY Theorem 1 . 6 , we h a v e
A l s o , i f v E a V ( B ) n r ( A ) n r v ( B w ) , a n d v w E E W ( A ) n r , ( ~ ) n r W ( ~ v ) , t h e n i t i s e a s y t o show t h a t
v E M and w E N , where M and N a r e a s i n Lemma 2 . 2 .
T h e r e f o r e RT(A(T) ) n r T ( A ( T ) C J ( T ) , which
i m p l i e s t h a t T = Q (T) , by Theorem 1 . 6 .
F o r t h e g e n e r a l c a s e , l e t H b e as i n Lemma 2 . 3 . Then ,
a s i n t h e p r o o f o f Theorem 3 .1 . H i s a l o c a l i z a t i o n
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o f T o v e r t h e s e t L = { ( E i ) ; c s C R ( 0 ) , d r C s ( O ) 1 .
T h e r e f o r e H i s a N o e t h e r i a n r i n g . A s i n t h e f i r s t
p a r t o f t h e p r o o f , we h a v e H Q ( H I . T h e r e f o r e H
i s a q u o t i e n t r i n g o f T.
F o r t h e C o n v e r s e , we c a n u s e s i m i l a r a r g u m e n t s a s
t h o s e u s e d t o p r o v e c o n d i t i o n (iii) i n Theorem 3 . 1 ,
t o show t h a t R and S w i l l s a t i s f y t h e l e f t and r i g h t
Ore c o n d i t i o n s , a n d , RVS and S W R w i l l s a t i s f y t h e
l e f t a n d r i g h t Ore c o n d i t i o n s w i t h r e s p e c t t o C ( 0 ) R
and C S ( 0 ) . A l s o i f H i s a s i n t h e f i r s t p a r t o f
t h e p r o o f , t h e n Q ( T ) E H.
C o r o l l a r y 3.5. ( [GI, P r o p o s i t i o n 4 . 2 ) . L e t R b e a
r i n b t h a t h a s a N o e t h e r i a n f u l l q u o t i e n t r i n g , s a y Q .
Then , f o r any n a t u r a l number n , M n ( R ) h a s a f u l l
q u o t i e n t r i n g i s o m o r p h i c t o M ( Q ) . n
C o r o l l a r y , 3 .6. L e t R b e a r i n g t h a t h a s a
N o e t h e r i a n f u l l q u o t i e n t r i n g , s a y Q . Then , f o r
any n a t u r a l number n , Tn ( R ) , t h e r i n g o f n x n
u p p e r ( l o w e r ) t r i a n g u l a r m a t r i c e s o v e r R h a s a f u l l
q u o t i e n t r i n g i s o m o r p h i c t o T ( Q ) . n
Remarks 3 . 7 . ( i) Example ( 2 . 1 ) o f [z] , shows t h a t
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t h e r e e x i s t s a r i g h t N o e t h e r i a n r i n g R wh ich h a s
a r i g h t q u o t i e n t r i n g b u t M 2 ( R ) d o e s n o t . So t h e
a s s u m p t i o n i n Theorem 3 .4 . t h a t T i s N o e t h e r i a n ( l e f t
and r i g h t ) i s n e c e s s a r y .
( 2 ) A l s o example ( 2 . 2 ) o f [ 2 ] , shows t h a t t h e r e
e x i s t s a r i n g T = ') w i t h t h e f o l l o w i n g p r o p e r t i e s ( w S
(i) T i s N o e t h e r i a n ( l e f t and r i g h t ) ,
(iii) RVS i s n o t t o r s i o n - f r e e l
( i v ) Q ( R ) d o e s n o t e x i s t .
So t h e a s s u m p t i o n s i n Thereom 3 . 4 , t h a t RVS and S W R
a r e t o r s i o n - f r e e a r e a l s o n e c e s s a r y .
( 3 ) F i n a l l y , t h e f o l l o w i n g example h a s b e e n u s e d i n
R V [GI, we u s e i t h e r e t o c o n s t r u c t a r i n g T = (W s )
w i t h t h e f o l l o w i n g p r o p e r t i e s .
( i) T i s N o e t h e r i a n ( l e f t a n d r i g h t )
( ii) RVS and = W R a r e
(iii) Q ( R ) and Q(S) b
( i v ) N e i t h e r RVS n o r
w i t h r e s p e c t t o
Theorem 3 . 4 . , Q
t o r s i o n - f r e e
t h e x i s t
S W R s a t i s f y t h e Ore c o n d i t i o n s
CR(0 ) and C S ( 0 ) . Hence by
T) d o e s n o t e x i s t .
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The Example : L e t R b e t h e c o m m u t a t i v e N o e t h e r i a n
r i n g
R o v e r a f i e l d K . L e t S = R A(R). Then R = Q ( R ) ,
S # Q ( S ) . I n f a c t Q(S) = R $ Q ( K [ y ] ) . Moreove r
S i s N o e t h e r i a n a s an S-module a n d a s a n R-module.
S e t T = R S (S S ) . Then T i s N o e t h e r i a n ,
R'S and S'R
a r e t o r s i o n - f r e e . We c l a i m t h a t RSS d o e s n o t s a t i s f y
t h e Ore c o n d i t i o n s w i t h r e s p e c t t o C (0) and C S ( 0 ) , R
f o r s u p p o s e i t d o e s . Choose a n e l e m e n t d o f S
w h i c h i s r e g u l a r i n S b u t n o t i n v e r t i b l e i n S . Then
t h e r e e i x s t C E C ~ ( O ) , s e S s u c h t h a t c . 1 = s d . Bu t
c -1 E R c S . SO d i s i n v e r t i b l e i n S w h i c h i s a
c o n t r a d u c t i o n . T h e r e f o r e Q ( T ) d o e s n o t e x i s t .
ACKNOWLEDGEMENTS
The a u t h o r would l i k e t o t h a n k P r o f e s s o r A.W. G o l d i e
and D r . J .C. Robson f o r many h e l p f u l c o n v e r s a t i o n s
and f o r e n c o u r a g e m e n t d u r i n g t h e w r i t i n g o f t h i s
p a p e r .
REFERENCES
[L] S.A. AMITSUR, R i n g s o f q u o t i e n t s a n d M o r i t a
ContexU, J . A l g e b r a 17 ( 1 9 7 1 ) , 273-298.
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at 0
3:03
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201
4
A.W. CHATTERS, T h r e e e x a m p l e s c o n c e r n i n g t h e Ore
c o n d i t i o n i n N o e t h e r i a n r i n g s , P r o c . E d i n b u r g h
Math. Soc .
A.W. GOLDIE, The S t r u c t u . r e o f N o e t h e r i a n r i n g s ,
L e c t u r e s on R ings and Modu les , L e c t u r e N o t e s
i n Math . , Vol . 246, S p r i n g e r - V e r l a g .
C . R . HAJARNAVIS, On S m a l l ' s Theorem, J. London
Math. Soc . , ( 2 ) , 5 ( 1 9 7 2 ) , 596-600.
C. LANSKI, N i l s u b r i n g s o f G o l d i e r i n g s a r e
n i l p o t e n t , C a n d i a n J . Math. , ( 1 9 6 9 ) , 9 0 4 - 9 0 7 .
T.H. LENAGAN, A r t i n i a n i d e a l s i n N o t h e r i a n
r i n g s , P r o c . Amer. Math. S o c . , % ( 1 9 7 5 ) ,
499-500.
L.S. LEVY, T o r s i o n - f r e e and d i v i s i b l e modu les
o v e r n o n - i n t e g r a l d o m a i n s , C a n a d i a n J . Math . ,
1 5 (19631 , 132-151. - A . D . SANDS, R a d i c a l s and M o r i t a C o n t e x t s .
J . A l g e b r a , z ( 1 9 7 3 1 , 335-345.
L.W. SMALL, O r d e r s i n A r t i n i a n r i n g s , J .
A l g e b r a i ( l 9 6 6 ) , 13-41.
L.W. SMALL, O r d e r s i n A r t i n i a n r i n g s 11,
J. A l g e b r a ( 1 9 6 8 ) , 266-273.
J . T . ST'AFFORD, N o e t h e r i a n f u l l q u o t i e n t r i n g s .
Received: Apri l 1981
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