可 換 環 のplace
岡 田 和 敏
Places of commutative rings
Kazutoshi Okada
The genararization of the concept of valuations of fields to valuations of commutative rings have been done by M. Manis (cf. [ 1 ]) and N. Bourbaki (cf. [ 2 ]) . In this note, we gener-alize the concept of places of fields to places of commutative rings. We show that a valuation
(v , r) of a ring R and a place p of R determine one another , and show some results. Through-out of this note, a ring always means a commutative ring with the identitiy and the identity of it's overring is the same.
Definition : A place of a ring R is a homormorphic mapping p of a subring A of R onto a domain A , which satisfies the following two conditions.
(1) If x E R—A, then there exists an element y in A such that p(y)-=- 0, xy E A and p(xy) # 0.
(2) There exists an element x E A such that p(x) # 0.
Let Mp be the set of all elements of A such that p(x)= 0. Then, it is clear that Mp is a prime ideal of A, and A is isomorphic onto A/Mp. If .x E R—A, then we say p(x) = 00 , otherwise p(x)# 00 . A is called a place ring of p, Mp is called a prime ideal of p, and A is called a re-sidue ring of p. The place ring of p of R denote Rp.
Definition : Let p, p' are places of a ring R with residue rings A , A respectively, we say that p and p' are isomorphic if there exists an isomorphic mapping sb of A onto A such that
= Op.
It is obvious that p and p' are isomorphic if and only if Rp = 4, and Mp =4' . Therefore, Proposition 1 of [ 1 ] shows that a class of places which are isomorphic to a place p determines a valuation (v, r) of R such that (Rv, Pv)= (Rp, Mp), where (R,„ P„) is the valuation pair of (v , r) . Converesely, for a valuation (v, r) of R , let p be the canonical map of R0 onto R„/Mp, then p is the place of R such that (R„, Pv)= (Rp, Mt) .
If a valuation (v, r) of R and a place p of R determines one another, we say (v , r) is associated with p, it follows that
^ (x)<v (1) <=> p (x)=0 ^ (x)‘v (1) <=> p (x)÷00 ^ (x)>v (1) <=> p (x)=00
Definition : Let p and p' be places of a ring R. If Rp' g Rp and Mp g Mp' , then we say that p' is a specialization of p
Proposition 1 Let p and p' be places of a ring R with residue ring A , A respectively. Then, p' is a specialization of p if and only if there exists a place q of A such that p' = qp on Rp'.
Proof : Let 0 q = P(Rp') P1 = p I Rp' , and q = p' 1. Then q is a homormorphic mapping of A, onto A . For every element E E A—A q, there is an elements x in Rp - Rp' such that p(x) = , and there exists y E Rp' such that p' (y) = 0, xy E Rp and p' (xy) = 0. We set p(y) = , then q( rl )= 0, E E A, and q( E i )# 0. Hence q is a place of A such that p' = qp on Rp'. Converse is clear.
Proposition 2 Let S be a subring of a ring R and let A be an proper ideal of S, then there exists a place p of R such that Rp S, Mp 2 A . In paticular if A is prime ideal, then Mp fl S
=A.
Proof : A may be assumed prime ideal. Let M = I (Ri, Pi)} iE/ be the set of pairs (Ri, Pi) ; Ri is a subring of R containing S and Pi is a prime ideal of Ri such that Pi fl S =A . It is clear that M is not empty. We define the order in M ; (Ri, Pi) z (Ri , Pi) if and only if Ri Ri and Pi 2 P1. In this order, M is inductive, therefore, M has a maximal pair (L, P) in M by Zorn's lemma. If there exsists a ring L' and a prime ideal P' of L' such that P' fl L=P, L' D L, then P' fl S= (P' fl L) fl S=A; this is the contradiction. Therefore, the maximal pair (L, P) satisfies the conditin (2) of Proposition 1 of [1]. It follows that (L, P) is a valuation pair. It is clear that the place associated with the valuation pair (L, P) satisfies the condition of the
Proposition.
Proposition 3 Let S be a subring of a ring R and let P and Q are prime ideals of S such that P g Q. If p is a place of R such that Mp fl S=P, then there exists a place q of R such that Mq
S=Q and q is a specialization of p.
Proof : There exists a place r of Rp/Mp, the place ring of r contains the subring S/P of Rp/Mp and the restriction of the prime ideal of r to S/P is Q/P. Let q=rp, then q is a place of R such that Rq S, Mq fl S=Q. It is obvious that q is a specialization of p.
Proposition 4 Let R be a ring, and let p be a place of R. If ai,••• ,a„, are elements of R such that for some ai, aiR C M. Then there exists an element a in R such that p(aia)+ co for
every i, and p(aja)* 0 for some j.
Proof : Let N be the set of elements x of R such thch xR g Mp. Let v be a vauation of R associated with p. If v(x) is not zero, then there exists y E R such that v(xy)=v (1). Since, v(x) = 0 if and only if x E N, we can take b, as an element of R such that v(aibi) =v(1) for each a1. Let a be one of the elements which take the least value among v(bi) . a satisfies the condition.
Definition : Let p be a place of R and p' is a place of a overring le of R. If p' IR is equal to p, we say p is an extention of p to R .
Propositin 5 immediately follows.
Proposition 5 Let p be a place of a ring R with residue ring A and let p be an extention of p to an overring le of R with residue ring A . If lx E R ; xR C Mpl =101, and if xi, •••, xm
E Rp• are lineary dependent over R. Let e P(xi)(i = 1, •••, m). Then, $1, •••, E. are lineary dependent over A .
References
[ 1 ] M. Manis, Valuations on a commutative ring, Proc. Amer. Math. Soc. , 20, (1969) 193-198.
[ 2 ] N. Bourbaki, Algebre commutative, 6, Hermann, (1964).
