Vol. XII, 1961 425

A N o t e o n t h e ' P r i m b a s i s s a t z "

By

SL P~v~x~r MVR~Y*)

I n [1] it was proved t h a t if A is a semilocal Cohen-Macaulay ring and p a prime ideal of r ank s contained in the radical such tha t A~ is regular, then there exist ele- ments al . . . . . as e p such t h a t rank (al . . . . . as) = s and (al . . . . . as), = pA~. The aim of this note is to prove tha t this result is t rue for any commuta t ive noetherian ring A with uni t element and any prime ideal p such tha t A~ is regular. I n fact we shall prove the following slightly more general result.

Theorem. Let A be a commutative noetherian ring with unit element and p a prime ideal o / r a n k s. Let q be a p-primary ideal and let n be the dimension o / g A ~ / q o A ~ as a vector space over Av /oAv . Then there exist elements al . . . . . an e g such that (al . . . . . an)~ = ---- qA~ and rank (al . . . . . as) = s. Further the a~ can be chosen [rom a min imal set o/

generators ]or q.

We first prove two lemmas. I n what follows A always stands for a commutat ive noether ian ring with unit element.

Lemma 1. Let Pl . . . . . Pl be pr ime ideals o / A such that p~ ~ 0 i /or i ee j . Let Yl . . . . . Yr

be elements o / A such that

(I) Y l e P ~ / o r s o m e i , l ~ i ~ = l , l

(II) (y~ . . . . . yr) CUt~i- ]=1

Then there exist elements a2, .. . , ar in A such that l

Yl + a2 y2 § "'" + ar Yr ~ U 01" 1=1

P r o o f . Choose xi e Pi such tha t xi ~ ~1 for i ~:j. Bevause of (II) for every Oi there exists a Y,(o, 1 ~ v(i) ~= l, such tha t Y~(O ~ P~, i = 1 . . . . . 1. We m a y assume yl e p~, 1 ~ _ i ~ t , a n d y ~ p j , t § Set

l :v i I t is immediate t h a t

l

*) My thanks are due to Dr. C. S. SES~XDRI and Mr. C. P. RA~.~cJi .~I for helpful suggestions.

426 M . P . Mu~z~Y ARCH. MATH.

1

Corollary 1. Let x be an element in A not contained in ( '~ OJ and let y e A . Then there exists ~ e A such that ~x + y ~ pj /or any j . /=1

l

P r o o f . I f y 6 [ J pj, choose 2 = 0. Otherwise apply Lemma 1 to the ideal (y, x). ]=1

Corollary 2. The ideal a = (yl . . . . . yr) in Lemons 1 can be generated by r elements l

which do not lie in ~ J OJ. j = l

l

P r o o f . By Lemma 1 we m a y assume Yl 6 U W " a nd ).~, 2 ~ i <= r, such tha t l I = 1

~ Y i -~- yi ~[,.JPJ. Then a = (Yl. ~2Yz + Y2 . . . . . ~rYi -}- Yr), q.e .d. ] = 1

Let a be any ideal of A. We denote by n (a) the least integer d such tha t a can be generated by d elements.

Remark. The ideal a = (Yi . . . . . Yr) in Lemma 1 can be generated by ~ (a) elements l

not lying in ~.J l~j- 1 = 1

Observing that, in a noetherian ring the set of zero divisors is precisely the un ion of the maximal prime ideals associated to (0), we have

Corollary 3. Let a be an ideal containing a non-zero divisor. Then a can be generated by ~o~-zero divisors.

Leinma 2. Let p be a pr ime ideal in A o] rank s >= 1. Let q be a 19-primary ideal such that the dimension o / q A v / q o A ~ as a vector space over A v / o A ~ is equal to n. Then there exist al . . . . . an e q not contained in any o/ the minimal pr ime ideals associated to (0) such that (el . . . . . an)v = qA~. These a~ can be chosen ]rom a minimal set o/generators o] q containing n (q) elements.

P r o o f . Let Oi . . . . . Ot be the min imal pr ime ideals associated to (0). As rank q l

1, we have q ~= [.J P3". Therefore by the remark after Corollary 2 there exist l ~=1

a i a q, al ~ [ J V j , 1 _< i _<_ n(q), such tha t q = (el . . . . . an(q)). 9 = 1

I t is well known tha t for a module of finite type M over a local ring A, elements Xl . . . . , xn e M form a min imal set of generators for M if and only if xi mod m M. 1 ~ i ~ n, form a basis for the vector space M / m M over A / m where m is the max imal ideal of A. 5Tow as the dimension of the vector space q A , / q p A , over A~/pA~ is n, we can choose n elements among the ai say ai . . . . . a n with (ai . . . . . an)~ = qAv , q .e .d .

We shall now prove the theorem by induct ion on s.

P r o o f of t h e T h e o r e m . Let p be a prime ideal d r a n k s and q a p-pr imary ideal such tha t the dimension of the vector space q A , / q a A v is equal to n. Let s = 0. Choose elements a i , . . . , a n (,) to generate q. Among these a~ we can choose a i , . . . , an

Vol. XII, 1961 A Note on the 'Primbasissatz' 427

such t h a t (al . . . . . an)~ = qA~ and r ank (a l , . . . , an) = 0. Le t s > 0. Assume tha t for all p r ime ideals p of r a n k d < s and for all p -p r imary ideals q there exis t e lements a i , . . . , an(a) which genera te q and f rom the ai we can choose e lements a l . . . . , am �9 q such t h a t (ai . . . . . am)~ = q A , and r ank (al . . . . . am) ----- d where m is the d imension of the vec to r space qA~/p qA~ over A p / ~ A ~ . B y L e m m a 2, we can choose e lements a i , . . . , an(q) genera t ing q ar.d no t in a n y min imal pr ime ideal associa ted to (0), such t h a t (al . . . . . an) , = qA~. Consider the r ing A ----- A/ (a l ) . We shal l denote the image in A of a n y idea l a ( respect ively an e lement x) of A b y -~ ( respect ively ~). We have A~ ---- A v / a l A v . As a l is no t conta ined in any of the min ima l p r ime ideals assoc ia ted to (0), r a n k ~ = d i m . 4 ~ = s - 1. The e lements a l . . . . . an m o d qpAv form a basis for q A v / q p A v as a vec tor space over A ~ / p A v a n d we have.

d im ( ~ A ~ f ~ A ~ ) = d im ( q A~I ~ ~ A~ + al A~) -~ n - - 1. (A~/~A~) (A~/I)Ap)

Therefore b y induc t ion hypothes is there exis t x l , . . . , Xn(~) �9 A such t h a t ~1 . . . . . xn(~) genera te ~ and (xz, - - - , xn-1)~ = q - ~ wi th r a n k (xl . . . . , Xn-1) = s - - 1. Hence (al , xz . . . . . Xn-1)~ ~- q A v and b y our choice of a l , r ank (a l , x l . . . . . xn-1)- - - -s . Since n(-q) ~ n(q) - - 1 and az. xz . . . . . xn(~) genera te q. Hence az, xz . . . . xn(~) is a min ima l set of genera tors for q and n(q) -~ n(q) - - 1.

Since in a local r ing of d imension s there exists a p r i m a r y ideal genera ted b y s ele- men t s a n d associa ted to the m a x i m a l ideal (see [2]), we i m m e d i a t e l y have

Corollary 1. I f A and p are as in the theorem there exist s elements a~ . . . . . as �9 p

such that r ank (az, . . . , as) = s and (az . . . . . as)~ is O A , pr imary .

I n the theorem if we t ake q ---- O and A v regnalar we get (see [1])

Corollary 2. Let Ap be regular. T h e n there exist elements al , . . . , as ~ p such that

r a n k (az . . . . . as) ~ s and (az . . . . . as)o = p A ~ .

Fur ther the a~ can be c h o s e n / t o m a m i n i m a l set o] generators o / ~ .

R e m a r k . I f the r ing A in the theorem is Cohen-Macaulay, then the a~ form an A-sequence (see [2]).

W e recal l (see [2]) t h a t in a c o m m u t a t i v e noe ther ian r ing A the e lements a~ . . . . . as �9 A form an A-sequence if ai+z is not a zero divisor in A / ( a z . . . . . ai), i ~ O, 1 . . . . .

s - - 1, (we t a k e a0 = 0). I t can be p roved t h a t in a local r ing A all the ma x ima l A-sequences have the same length and t h a t th is length cannot exceed the d imension of the local r ing A. I f the l e n ~ h of a n y m a x i m a l A-sequence is equal to the d imension of the local r ing A, we say t h a t A is Cohen-Macaulay . We say t h a t a c o m m u t a t i v e noe the r ian r ing A is Cohen-Macaulay i f each A m is Cohen-Macaulay of the same d imens ion for every m a x i m a l idea l m of A.

W e have the following s t rengthening of Pr imbas i ssa tz (see [2]) in case A~ is reg- ular.

Corollary 3. Under the hypothesis o] Corollary 2, there exist elements a~ . . . . . as �9 p

such that r ank (a~, . . . , as) = s and p = (az, . . . : as) : t wi th t ~ p.

428 M.P. Mua~aY ARCH. MATH.

P r o o f . L e t b l . . . . . bm generate ~. Choose the elements a l . . . . . as by Corollary 2. Since (al . . . . . as)p ---- pAp, there exist t~ ~ p such tha t tib~ e (a l . . . . . as) , 1 <_ i <-- m .

Set t ~ ] -~ t i . Then p = (al . . . . . as) : t.

Relerenees

[1] H. J. NASTOLD, ZumPrimbasissatzinregul~renlokalenRingen. Arch.Math. 12,30--33(1961). [2] J. P. Sv.m~v,, Alg~bre Locale -- )r Cours au Coll~ge de France 1957--1958.

Anschrift des Autors: M. Pavaman Murthy School of Mathematics Tara Institute of Fundamental Research Colaba Bomb~- 5, India

Eingegangen am 22.8. 1961