数理解析研究所講究録 1222 巻 2001 年 18-23

18
The ideal transforms of semigroups
愛知教育大学金光三男 (Mitsuo Kanemitsu)

Aichi University of Education

By asemigroup we mean asubmonoid of atorsion-free abelian (additive) group in this paper. Let be asemigroup with the quotient group$q(S)$ , that is, $q(S)=\{s -s’|s, s’\in S\}$ . Any semigroupbetweenand $q(S)$ is called an oversemigroup of .
Moreover, let be the set of all integers and let. And $S[X]=S+\mathrm{Z}\mathrm{o}X=\{s$ $+nX|s\in$
and $X$ .
,is called apolynomial semigroup over (cf. [KOM]).
Let I be asubset of S. I is called an ideal of if $+S=I$ , thatis, $a+s$ $\in I$ for each $a\in I$ and each $\in S$ . For any $a\in S$ , put$(a)=a+S=\{a+s |s \in S\}$ . Then (a) is an ideal of and, $a_{n}\in S$ , we set
it is called aprincipal ideal of . For , a2,, ). The ( , a2,,
andis an ideal of and it is called an ideal generated by , a2,$\{a_{1}, a_{2}, \cdots, a_{n}\}$ is called a basis of.
An element of is called a unit if $+v=0$ for some $v\in S$ . Let$U(S)$ be the set of units in. Note that $U(S)$ be asubgroup of $q(S)$ .
If we put $M=S-U(S)$ , then $M$ is an ideal of . Moreover, if I is
, then $M=I$ or $=S$ . $M$ is called thean ideal of such that $M$
masirnal ideal of . Aproper ideal of is cffied a prirne ideal of if$a+b\in P$ with , $b\in S$ implies either $a\in P$ or $b\in P$ . We note that the
maximal ideal of is aprime ideal, is aprime ideal of and has the
only one maximal ideal.
$S$

$T$

$S$

$S$

$\mathrm{Z}_{\mathrm{n}}=\{a\in \mathrm{Z}|a\geq n\}$$\mathrm{Z}$$\mathrm{Z}_{\mathrm{m}}=\{aX|a\in \mathrm{Z}_{\varpi}\}$

$S$

$S$

$n\in \mathrm{Z}_{0}\}$

$S$

$I$

$s$

$S$

$S$

$I$ $=$$\cdots$$a_{1}$$( \mathrm{a}\mathrm{x}, a_{2}, \cdots, a_{n})=\bigcup_{\dot{|}=1}^{\mathfrak{n}}(\alpha.)=\mathrm{u}_{=1}^{\mathfrak{n}}.\cdot(a_{t}+S)$$\cdots$$a_{1}$$S$$\cdots$$a_{1}$$a_{n}$$a_{n}$

$I$

$S$

$u$

$u$

$S$

$S$

$S$

$I$

$\subset I$

$S$

$P$

$S$

$\phi$

$S$

$S$

$a$

$S$

$S$

We give semigroup versions of some results in [F].

be asemigroup. Also, letof . For an ideal I of , we put $V(I)$
Let
$S$

$S$

$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(S)$$S$

be the set of all prime ideals
and. In particular, put$=\{P\in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(S)|P\supset I \}$$D(I)$ $=\{P\in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(S)|P\not\supset I \}=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(S)-V(I)$$D((a))=D(a)$ for $a\in S$ .
is a family of ideals of and let I andLemma 1. Let
ideals of S. Then we have the following statements$\{I_{\lambda}\}_{\lambda\in\Lambda}$

$S$

$J$

are

19
(1)
is an ideal of $S$ .
(2) {A} is an ideal of $S$ .
(3) $I+J=\{a_{l}+b|a\in I, b\in J.\}$ is an ideal of $S$ such that $I+J\subset I\cap J$ .
(4) If $P=I$
is a prime ideal of $S$ , then $=P$ or $J=P$ .
(5) If $P$ and $Q$ are two prime ideals of $S$ , then $P\cup Q$ is also a prime$\cap\{I_{\lambda}|\lambda\in\Lambda\}$$\cup$$\in\Lambda$$I_{\lambda}|$$\cap J$

ideal of

$S$

$I$

.

Lemma 2. Let$S$

be a semigroup. Then the following statements hold.

(1), $V(S)=\phi$ .
(2) If, then $V(I)$.
(3) $V(I_{1})\cap V(I_{2})=V(I_{1})\cup V(I_{2})$ .
(4) $V$ (A) $=\cap$ {A$V(\phi)=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(S)$$I$$\subset J$$\supset \mathrm{V}\{\mathrm{S})$$\cup\{I_{\lambda}|$$\in\Lambda\}$$V(I_{\lambda})|$$\in\Lambda$}.

We makeinto atopological space; the topology is called theZariski topology. The closed sets are defined by the $V(I)$ $=\{P\in$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(S)$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(S)|P\supset I\}$.

Then $D(I)$ is an open set of$f\in P\}$ is an open basis offollowing statement.

$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(5)$Proposition 3.
quasi-compact space.

$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(S)$and the. For this topology, we give the$D(f)=\{P\in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(S)|$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(S)$isa Kolmogoroff space (Definition 1. We call the ideal transfor$m$ of
I of the following oversemigroup of :
$S$

$S$

$T_{0}$

space; and$a$with respect to an ideal$S$$T_{s}(I)$

$:=$

{z

$\in q(S)$|(S:sz $+S)\supset nI$ for some n$\geq 1$}

Also, we call the Kaplansky ideal transform of S with respect to an ideal
I of S the following oversemigroup of S:$\Omega s(I)$

$:=$

{z

$\in q(S)$| rad(5 :sz$+S)\supset I$}.

where rad(J) { $a\in S|na\in J$ for some positive integer }.
Note thatis an oversemigroup of
and note.that if I is$=T_{S}(I)$ . For aprincipal ideal
finitely generated, then, we havethat $+T_{S}(I)$ $=T_{S}(I)$ .
$=$

$n$

$\Omega_{S}(I)$$T_{S}(I)$$\Omega_{S}(I)$

$I$

$I$

20
Proposition 4. Let I be a princiapal ideal of and $P$ be a prime idealof S. Then the following results are hold.
.
(1) $+T_{S}(I)$ $=T_{S}(I)$ ,(2) $P\in V(I)$ if and only if $P+T_{S}(I)$ $=T_{S}(I)$ if and only if $P+\Omega_{S}(I)=$$S$$I$ $+\Omega_{S}(I)$$I$$\Omega_{S}(I)$$=\Omega_{S}(I)$.

Definition 2([K],[KB],[KM] and [MK]). Asemigroup is avaluation.
thensemigroup if, $3\alpha\in S$ foris aseminormal semigroup if
Also, we say that.
, we have$S$$\alpha\in S\mathrm{o}\mathrm{r}-\alpha\in S$$\alpha\in q(S)$$2\alpha$$S$$\alpha\in S$$\alpha\in q(S)$It is clear that valuation semigroups are seminormal.

Definition 3. Anon-empty subset $N$ of asemigroup is called an
additive system of if , $b\in N$ implies $a+b\in N$ and $0\in N$ .
is an oversemigroup of
Put $S_{N}=\{s-t |s\in S, t\in N\}$ . Thenand is called the quotient semigroup of . If is aprime ideal of , thenis
$T=S-P$ is an additive system of and the quotient semigroup
denoted by .
$S$

$S$

$a$

$S$

$S_{N}$

$S$

$P$

$S$

$S_{T}$$S$$S_{P}$

Definition 4([OK]). Let be an oversemigroup of . Then is saidof , either $P+T=T$ or
to be flat over if for any prime idealor}.
. Put Flat(T) {
$S$

$T$

$=$

$T\subset S_{P}$

$S$

$P$

$S$

$T$

$P\in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(S)|P+T=T$$T\subset S_{P}$Example 1. Let $S=(\mathrm{Z}_{1}+\mathrm{Z}_{1}X)\cup\{0\}$ . Then $U(S)=\{0\}$ andis the maximal ideal of . Also, Krull $\dim$$S=1$ andis not valuation semigroup.
. Then is not avaluation semigroupAlso, let.
. Thenand $U(T)=\{0\}$ . Put .
$S$$M=\mathrm{Z}_{1}+\mathrm{Z}_{1}X=S-\mathrm{C}/(\mathrm{S})$

$S$

$T$

$T=(\mathrm{Z}_{1}+\mathrm{Z}_{1}X)\cup \mathrm{Z}_{0}$$N\not\in \mathrm{F}1\mathrm{a}\mathrm{t}(\mathrm{T})$$N=\mathrm{Z}_{1}+\mathrm{Z}_{1}X$

Theorem 5([OK]). Let be an oversemigroup of S. Then the follow$ing$ statements are equivalent$T$(1) $T$ is flat over $S$ .
(2) $T=S_{N\cap S}$ for the maximal ideal $N$ of .
(3) For any two idealS ,of , $(I \cap J)+T=(I +T)\cap(J+T)$ .
$T$

$J$

$I$

$S$

Definition 5. Let be asemigroup and let be an oversemigroup of. Then is said to be $LCM$-stable over if $((a+S)\cap(b+S))+T=$$(a+T)\cap(b+T)$ for each , $b\in S$ .
$T$

$S$

$S$

$S$

$T$

$a$

21
Aflat oversemigroup T over S is LCM-stable over S.
Theorem 6. Assume that be a Noetherian semigroup. Let be anoversemigroup of S. Then is flat over if and only if is LCM-stable
over .
$S$

$T$

$S$

$T$

$T$

$S$

Corollaly 7. If is a valuation semigroup and a proper principal ideal$=(a)$ of , then $P\in D(I)$ if and only if $T_{S}(I)$
.
$S$

$I$

$S$

$=\Omega_{S}(I)$$\subset S_{P}$Proposition 8. The following satetements are hold.
(1) $S_{a}=\Omega_{S}((a))$ for a non-unit $a\in S$ .
(2) If I and are ideals of such that, then$J$$S$$I$ $\subset J$$\Omega_{S}(I)$$\supset\Omega_{S}(J)$.

$=\cap\{S_{P}|P\in D(I)\}=\cap\{\Omega_{S}(I+S_{P})|P\in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(5)\}$.
(3)$=\cap\{\Omega_{S}(a+S)|a\in I\}$
(4) If I is a proper ideal of $S$ , then$\Omega_{S}(I)$$\Omega_{S}(I)$$\cap\{S_{a}|a\in S_{a}\}$$=$.

Definition 6. $\in G$ is called an almost integral element of if there
exisits an element $a\in S$ such that $a+nx$ $\in S$ for each positive integer. Also, is acompletely integrally closed if is almost integral overthen $x\in S$ .
$S$

$x$

$n$

$S$

$S$

$x$

Theorem 9([K]). $Lei$ be a valuation semigroup such that $S\neq q(S)$ .
Then Krull $\dim S=1$ if and only if is a completely integrally closed
$S$

$S$

semigroup.

(1) $, (1, X), \phi\}$ .
Theorem 10 ([KHF]). (1)(2) The primary ideals ofare the following:(i) All the ideals that contains both elements of
.
and
(ii)
.
with
(ii)
.
with
$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathrm{Z}_{0}[X])=\{(X),$$\mathrm{Z}_{0}[X]$$\mathrm{Z}_{0}$$\mathrm{Z}_{0}X$$k\in \mathrm{Z}_{0}$$\mathrm{Z}_{\mathrm{k}}+\mathrm{Z}_{0}X=(k)$$\mathrm{Z}_{0}+\mathrm{Z}_{\mathrm{k}}X=(\mathrm{h}\mathrm{i})$$k$$\in \mathrm{Z}_{0}$Example 2. Let. Put
and $M=$(1) $=1$ $+S=P\cup \mathrm{Z}_{1}$ . Then,
. Since
we have that Krull $\dim S=2$ . It is clear that is avaluation semigroup.
Since $P$ is not finitely generated, is not Noetherian semigroup.
$S=\mathrm{Z}_{0}\cup(\mathrm{Z}+\mathrm{Z}_{1}X)$$P=\mathrm{Z}+\mathrm{Z}_{1}\mathrm{X}$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(S)=\{\phi, P, M\}$$\phi\subset P\subset M$

$S$

$S$

Theorem 11. Let the notation be as in Example 2and let I be an
ideal of S. Then the following statements hold22
be a principal ideal of S. If $f=0$ , then $T_{S}(I)=$$=S$ . Also, if $f\in M-P$ , then$=S_{f}=\mathrm{Z}[X]=$. Next, if $f\in P$ , then$=S_{f}=q(S)$ .
(2) If I is not a finitely generated ideal of , then$=q(S)$ .
and(3) Let. Thenif and only if(1) Let$I$$=(f)$$\Omega s(I)$$T_{S}(I)$$=\Omega_{S}(I)$$T_{S}(I)$ $=\Omega_{S}(I)$$\mathrm{Z}+\mathrm{Z}_{0}X$$S$$I$ $=\mathrm{Z}+\mathrm{Z}_{\mathrm{n}}X(n\geq 1)$$\Omega_{S}(I)$$I$$\Omega_{S}(I)$$\neq S$

$I$ $+\Omega_{S}(I)$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\Omega_{S}(I))\cong D(I)$$=$.

(4) $S$ is not a completely integrally closed and each oversemigroup of $S$
is a flat semigroup over $S$ , and so Flat(T)for each overserni$=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(S)$

group

$T$

of .
$S$

Theorem 12. Let be a semigroup and I an ideal of S. Then thefollowing statements are equivalent.
(1)
is an affine open subspace of(2)is flat over and, for each
, $P+$
with$S$$D(\mathit{1})$$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathrm{S})$$S$$\Omega_{S}(I)$$\Omega_{S}(I)$(3)$=\Omega_{S}(I)$.

$I$ $+\Omega_{S}(I)$$=\Omega_{S}(I)$$P\in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(S)$$P\supset I$.
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[KM] M.Kanemitsu and R Matsuda, Note on seminormal overrings, Hoetston J. Math. 22 (1996), 217-224

23[KOM] M.Kanemitsu, A.Okabe and R.Matsuda, On polynomial extensions of -monoids, Algebras Grryups and Geometries 16 (1999), 269$\mathrm{g}$-276.

[MK] R Matsuda and M.Kanemitsu, On seminormal semigroups, Arch.
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