Arch. Math., Vol. 37, 325--329 ( 1 9 8 1 ) 0003-889X/81/3704-0005 $ 01.50 + 0.20/0 �9 1981 Birkh~iuser Verlag, Basel

A note on ideals in polynomial rings

By

l ~ o D K. S ~ - ~

Introduction. All rings are commutative with identity. Let R [X] be a polynomial ring over an integral domain. Kaplansky (.[3], 102, Exercises 3 and 4) has shown that for a, b e R, the ideal ( a X + b) is prime in R [X] ff and only a, b is an R- sequence. We shall give an interesting criterion (Theorem 1) for a prime ideal P in R [X] with P c~ R = (0) to be generated by an irreducible polynomial. I t gen- eralises the result in Kaplansky [3] and also a classical result due to Gauss which states that R IX] is factorial ff R is factorial if we assume that a prime element in R is prime in R [X]. We then deduce affirmative results for unique factorisation domains and valuation rings. We also prove results (Lemma 1 and Theorem 2) giving conditions under which a prime ideal in R [X] is invertible.

1. We prove the following:

Theorem 1. Let R [ X] be a polynomial ring over an integral domain R and let P (:4=0) be a prime ideal in /~[X] such that P N R = (0). Let

q)(X) = aoX a + al X a-i + ... --~ ad-i X -~- aa

be a polynomial o/least positive degree in P. Then P ---- (q)(X)) i / a n d only i / there does not exist t ~ (ao) such that tai E (a0) /or 1 <_ i ~ d.

P r o o f . Let us assume P = (~(X)). Suppose there exists t ~ (a0) such that ta i~ (ao) for 1 ~ i _~ d. We will arrive at a contradiction. Let

ta~=o~ao, i---- 1,2, . . . , d ; a l ~ R .

Then

ao(tX a + ~ I X d-1 "J- " '" -~ ~a) = t ~9 (X) ~ P .

This implies that

t X a + o:lX a-1 -}- "" + : ta- lX + ~a e P ,

since P is prime and P c~ R = (0). Then since the degree of ~ (X) is d, there exists 2 E R such that

~ ( X ) = t X a + ~ l X a-1 + ... + ~a.

326 P.K. SHA~MA ARCH. MATH.

Thus t ~ ~a0 e (a0), a contradiction to the assumption that t ~ (a0). Conversely, let the condition in the theorem hold. We shall show that P---- (~0(X)). Take any element g ( X ) e P. Since ~(X) has an invertible leading coefficient in B[1/ao][X], we can find h (X), r (X) in R IX] such that

(i) a~g(X) --- of(X)h(X) + r (X) ,

where deg. r (X) <~ d. Now from (i), we get

r (X) = a~ n g (X) -- ~ (X) h (X) e P .

This implies

r(X) = 0

since deg. r (X) ~ d and P (~ R = (0). Therefore

a~g (X) = (p (X) h (X).

Choose iV least such that

(ii) alCo g(X) --- c f (X)h(X) .

I f iV ---- 0, we get g(X) e (~(X)) . Suppose N > 0. Consider the natural map R [x ] -+ •/(ao) EX].

In R/(ao)IX], (ii) yields

(z) ~(z) = o,

where ~ (X), ~(X) are images of ~ (X) and h(X) respectively in .R/(ao)IX]. Now, if ~ (X) = 0 then clearly every coefficient of h (X) is a multiple of a0 in R. Thus from (ii) we get

aft -1 g (X) = ~o (X) h'(X),

where h'(X) is obtained by dividing coefficients of h(X) by a0. This contradicts minimality of N. Hence ~(X) :4: 0. Thus ~(X) is zero divisor in R/(ao)[X]. Hence there exists i (4=0) in B/(ao) such that iF (X ) -----0. This shows that there exists t ~ (a0) such that ta~ e (ao) for all i ~ 1. A contradiction to the assumption. There- fore g(X) ~- h(X) q)(X) and hence P ---- (~(X)). Thus the theorem follows.

Corollary 1. Let R be a unique/actorization domain or a valuation ring and P ( ~ O) a prime ideal in R IX] with P n R ~ (0). Then there exists an. irreducible polynomial ~(X) with least positive degree among elements ( ~, 0) o/ P such that P : (q~(X)).

P r o o f . Choose ~ ( X ) e P as in the theorem such that the greatest common di- visor of the coefficients a0, al . . . . , aa of ~ (X) is 1. Let there exist t ~ (ao) such that t a~ ~ (ao) for all i ~ 1.

Now, consider the two eases separately:

(i) B: Unique factorisation domain.

Let

ta~--2tao, i ~ 1 , 2 , . . . , d .

Vol. 37, 1981 Ideals in polynomial rings 327

This implies t g.c.d. (al . . . . , an) = a0 g.c.d. (~1, . . . , 2~). Hence a0 divides t since g.c.d. (a0, (al, . . . , a~)) ~ 1. A contradiction to the assumption tha t t q~ (a0).

(ii) R: Valuation ring.

In a valuation ring for any two ideals (a), (b) either (a) v (b) or (b)~ (a). There- fore assume (aio) ~ (a~) for every k = 0, 1, . . . , d. This implies tha t ai0 is a factor of each al, 7" = 0, 1 . . . . . d. Consequently aio is a unit since (a0, a l , . . . , an) ---- 1. Thus it is clear tha t there is not t ~ (a0) such tha t ta~ = ~tao for all i -= 1, 2 . . . . , d.

Corollary 2. Let R be an integral domain and let a, b be two non.zero elements o / R . Then (aX ~ b) is a prime ideal in R[X] if and only i /a , b is an R-sequence.

P r o o f . I f (aX ~ b) -= P is a prime ideal, the proof is immediate by the theorem. Conversely let a, b be an R-sequence. I t i s clear tha t a X ~ b generates a prime ideal Q in K[X]. Let

P = Q n R IX] .

Then clearly P is a prime ideal in R[X] containing a X -k- b such that P(~ R = (0). The rest is immediate.

Let us recollect tha t the content of a polynomial ~ (X) in R [X] i.e. cont. ~ (X) is the ideal of R generated by the coefficients of fr (X).

Corollary 3. Let R be an integral domain and I (~=0) an ideal in R[X] such that I ~ R ----- (0). I / there exists a polynomial qv (X) e I such that q) (X) is o/least positive degree in I and cont. (q~ (X)) ---- R, then I is generated by the polynomial q)(X).

P r o o f . Let ~v(X) ---- aoX ~ ~- a l X ~ --? ... ~- a~- lX ~- a~. Let t e R be such tha t t a~ e (a0) for all 1 --< i _< d, then clearly t e (a0) as the content of ~0(X) ---- R. Now, the result follows as in the theorem.

R e m a r k s . (i) The sequence a0, al . . . . . ad of the coefficients o f ~o(X) in the theorem need not be an R-sequence in general e.g. 2X 2 ~ 4 X A- 1 generates a prime ideal Q in Z IX] such tha t Q t3 2[ = (0), but, 2, 4, 1 is not a 7/-sequence.

(ii) The reverse s ta tement in the theorem is valid without the assumption tha~ P is a prime ideal.

(iii) The theorem is valid for any ring if we assume tha t ao is a non-zero-divisor.

2. We give below some conditions under which a lJrime ideal in R [X] is invertible as applications of the resu]ts proved in the 1. The lemma 1 and Cor. 4 are well known.

Lemma 1. Let R be a regular ring. Any prime ideal P o / R [X] such that P n R ~ (0) is invertible in R [X].

P r o o f . The result will be proved by showing tha t P(R[X])M is principal for every maximal ideal M of R[X]. Take any maximal ideal M in R[X], then M (3 R ~--Q is a prime ideal in R. We shall show tha t P RQ[X] is principal. I f

328 P.K. SHAP, MA ARCH. MAW.

Q -- (0) then RQ -~- K, the field of fractions of R. In this case the result is trivial. Let Q ~= 0. Then RQ is a unique factorization domain. Therefore the result follows from corollary 1. Thus P is invertible in R[X].

Corollary 4. Let R be a regular ring and let P be a prime ideal in R [X] such that P n R -~ (0). Then there exists an ideal A o / R such that P is isomorphic to A [X] as R [X]-module.

P r o o f . In the Lemma we have shown that PM is principal in I~M[X] for every maximal ideal M of R. Thus P is locally extended. The result now, follows from [2].

Theorem 2. Let R be a -~oetherian integral domain and let P ( 5 O) be a prime ideal in R IX] such that P n R : (0). I / there exists a polynomial q)(X) in P such that q) (X) is o/least positive degree in P and the content o/q)(X) is an invertible ideal o / R , then P is invertible in R IX].

P r o o f . Let r -~ aoX ~ ~ a i X ~-i ~ ... ~ a ~ - i X + a~ be of degree d. By as- sumption the ideal J ~ (a0, ai , . . . , a~) is an invertflsle ideal in /L Therefore for any prime ideal Q e Spec R, JRQ is a principal ideal. I t is easy to see that for any Q e Spec R, there exists h (X) e R [X], C e R, s e R -- Q such that

(x) o h(X) (i) 1 -- s 1

in RQ [X] and the content of h (X) is not contained in Q. Let I ' denote the ideal in R IX] generated by all such h (X). From (i), we get, in turn

8 (~) ~ . ~ (x) = h (X),

8 -~-" J ---- cont. h (X) c R ,

h (X) e j - i ~ (X),

I ' cJ- iq~(X) . R[X] .

Further, (ii) gives that for any Q e Spec R, there exists s e R - Q such that. s.q~(X) e I ' . Therefore it is immediate tha t ~ ( X ) ~ I ' . 1Wext note tha t

8 -~- J ---- cont. (h (X)) r Q.

Thus one can easily see that the elements s/C generate j - i . Now, since by (ii) we have that s /Cq) (X)e l ' , it is clear tha t I ' ~ J - i ~ ( X ) R [ X ] .

From (ii), sq~(X) ~ Ch(X) . Since ~(X) e P, it follows that Ch(X) e P. There- fore since P is prime ideal and Bf~ R ~- (0), we get h(x) e P. Consequently we get that for any Q e Spec R, there exists a polynomial h ( X ) e P such tha t the degree of h (X) is d and the content of h (X) is not contained in Q. Now, take Qi e Spec/~. There is a polynomial g(X) ~ P with deg. g(X) ---- d (least -{- ve among

Vol. 37, 1981 Ideals in polynomial rings 329

elements of P) and content of g(X) is not contained in QI. Thus we get tha t the image g'(X) of g(X) in PRetl[X] has least ~ ve degree among elements of PRQI[X ] and the content of g'(X) is R~,. Consequently by corollary 3, we conclude that PRc~I[X] is principal. Thus it is immediate tha t P is invertible in R[X].

R e m a r k s . (i) In the theorem even if we assume P be any ideal 1, we shall get / , = j - 1 ~ (x ) R IX].

(ii) The ideal P in the theorem is not generated by ~(X) unless the content J of ~(X) is equal to R. I f J ~ R, then clearly j - 1 4: R. Therefore the length of any maximal R-sequence contained in J is utmost 1 i.e. G(J) ~ 1. Now, since G(J) =< 1, i t is immediate tha t JcZ(R/(ao)). Thus there exists t ~ (ao) such that ].tc(ao). Now it is clear from the theorem 1, tha t P=~ (~0(X)).

These results developed in May, 1979, from a course of lectures in a summer school in Algebra for college lecturers. Thanks are due to Dr. T. Soundararajan with whom I had some interesting discussions.

References

[1] M. F. ATIYAH and I. G. MACDO~Ar.D, Introduction to Cvmmutative Algebra. 1969. [2] H: BAss, Torsion Free and Projective Modules. Trans. Amer. Math. Soe. 102, 319--327 (1962). [3] lmVlXCG K~I~SXV, Commutative Rings. Chicago 1974. [4] D. E. Rvsa, Pie R and R-Flatness of R [X]/I. J. Pure Applied Algebra 14, 307--310 (1979).

Anschrift des Autors:

Pramod K. Sharma Department of Mathematics University of Indore Vigyan Bhavan Khandawa Road Indore-452 001 India

Eingegangen am7.7. 1980")

*) Eine Neufassung ging am 24. 2. 1981 ein.