数理解析研究所講究録 1222 巻 2001 年 11-17

11
A Localization of a Semigroup Ring,II
松田 隆輝 (Ry\^uh. Matsuda)
茨城大学理学部 (Faculty of Science, Ibaraki University)

This is acontinuation of our [M4]. Thus asubmonoid of atorsion-freeabelian (additive) group is called a -monoid. For a -monoid , the quotientgroup of is denoted by $q(S)$ , and for acommutative ring , the total quotientring of is denoted by $q(R)$ . Throughout the paper denotes ag-monoid
which is not {0}.
Let $F(S)$ be the set of fractional ideals of the -monoid . Amappingof $F(S)$ to $F(S)$ is called astar-0peration on if the following
conditions hold for every element $a\in q(S)$ and ,:$S$$\mathrm{g}$$S$$\mathrm{g}$

$S$

$R$

$R$

$S$

$S$

$\mathrm{g}$$I$$\mapsto I^{*}$

$S$

$I$

$(a)’=(a);(a+I)^{*}=a+I^{*};I$$J\in \mathrm{F}(\mathrm{S})$$\subset I^{*};$, then.
astar-0peracion on . If, for all finitely generated fractional ideals’,
a1ld , (implies,called an e.a.b.
star-0peration on .
Let $F’(S)$ be the set of non-empty subsets of $q(S)$ such that $S+I$. A
mappingof
to $F’(S)$ is called asemistar-0peration on if thefollowing conditions hold for every element $a\in q(S)$ and , $J\in F’(S)$ :

If

$I$

$\subset J$$I^{*}\subset J^{*}j(I^{*})^{*}=I^{*}$$S$$\mathrm{L}\mathrm{e}\mathrm{t}*\mathrm{o}\mathrm{e}7$$J_{1}$$I$$J_{2}$$I$ $+J_{1J}^{\backslash *}\subset(I +J_{2})$$J_{1}^{*}\subset J_{2}^{*}$$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}*\mathrm{i}\mathrm{s}$$S$$\subset I$$I$$\mapsto I^{*}$$S$$F’(\mathrm{S}\grave{)}$$I$$(a+I)^{*}=a+I^{*};$ $I$$\subset I^{*};$$(I^{*})^{*}=I^{*}$ .
, thenasemistar-0peration on . If, for all finitely generated ffactional
ideals , and , $(I +J_{1})^{*}\subset(I +J_{2})^{*}$ implies,
called an
e.a.b.
star-0peration on .
$R$Letbe acommutative ring. Anon-zerodivisor of is called aregularelement of . If an ideal I of $R$ contains at least one regular element, thenI is called aregular ideal of . If every regular ideal is generated by regularelements, then $R$ is called aMarot ring. If, for every regular element $f$ of thepolynomial ring $R[X]$ , the ideal of generated by the coefficients of $f$ is aregular ideal of $R$ , then $R$ is said to have property (A).
Let I be an $R$-submodule of $q(R)$ such that $rI$
for some regular $r\in R$.
Then I is called afractional ideal of $R$ . Let $F(R)$ be the set of non-zero
fractional ideals of $R$. Amappingof $F(R)$ to $F(R)$ is called a

If

$I$

$\subset J$$I^{*}\subset J^{*};$$S$$\mathrm{L}\mathrm{e}\mathrm{t}*\mathrm{b}\mathrm{e}$$J_{1}$$J_{2}$$I$$J_{1}^{*}\subset J_{2}^{*}$

$S$

$R$

$R$

$R$

$R$

$\subset R$$I$$\mapsto I^{*}$$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}*\mathrm{i}\mathrm{s}$12star-0peration on if the following conditions hold for every regular element$a\in q(R)$ and , $J\in F(R)$ :
$R$

$I$

(1) $(a)’=(a);(2)(aI)^{*}=aI^{*};(3)I$(5) $(I^{*})^{*}=I^{*}$ .
, then(4) Ifastar-0peration on $R$ . If, for all finitely generated non-zero fracimplies,
tional ideals , , I with I regular,
is called an e.a.b. star-0peration on $R$ .
Let $F’(R)$ be the set of non-zero $R$-submodules of $q(R)$ . Amappingof $F’(R)$ to $F’(R)$ is called asemistar-0peration on $R$ if the following conditions$\subset I^{*};$$I$$\subset J$$I^{*}\subset J^{*};$$\mathrm{L}\mathrm{e}\mathrm{t}*\mathrm{b}\mathrm{e}$$(IJ_{1})^{*}\subset(IJ_{2})^{*}$$J_{2}$$J_{1}$$J_{1}^{*}\subset \mathrm{J}2$$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}*$$I$ $\mapsto I^{*}$

hold for every regular element $a\in q(R)$ and , $J\in F’(R)$ :$I$(1) (1)(3) If$=aI^{*};$

(2)

$I$

$\subset I^{*};$, then $I^{*}\subset J^{*};(4)(I^{*})^{*}=I^{*}$ .
a semistar-0peration on $R$ . If, for all finitely generated non-zeroLetfractional ideals , , I with I regular,, thenimpliesstar-0peration on $R$ .
called an e.a.b.
Let $f= \sum_{1}^{n}a_{\dot{l}}X^{s:}\in R[X;S]$ , where
for $i\neq j$ , and .
for each .
of is denoted by $e(f)$ , and the ideal
Then the ideal
of $R$ is denoted by $c(f)$ .
$I$$\subset J$$*\mathrm{b}\mathrm{e}$$J_{1}$$(IJ_{1})^{*}\subset(IJ_{2})^{*}$$J_{2}$$J_{1}^{*}\subset J_{2}^{*}$$*\mathrm{i}\mathrm{s}$$s_{\dot{1}}$$\neq s_{j}$$\alpha$$S$$(s_{1}, \cdots s_{n})$$\neq 0$$i$$(a_{1}, \cdots, a_{n})$Proposition 1. Let
astar-0peration on adomain $D$ . The following
conditions are:(1)
e.a.b.
(2) If $f/g=f’/g’$ , where $f,g$ , $f’,g’\in D[X]$ with $g,g’$ non-zero, and if, then.
$*\mathrm{b}\mathrm{e}$$\Re \mathrm{u}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{t}$$*\mathrm{i}\mathrm{s}$$\mathrm{c}(f)^{*}\subset c(g)^{*}$$\mathrm{c}(f’)^{*}\subset c(\oint)^{*}$’
Proof. Assume that, if $f/g=f’/g’$ , where$f’,g’\in D[X]$ with
non-zero, and if, then. Let , , be finitely
generated non-zero ffactional ideals of $D$ , and assume that.
We may assume that , ,are ideals of $D$ . Let(,an), $J_{1}=$and. Put $f= \sum a:X^{:},g=\sum b:X^{\dot{\iota}(n+1)}$ md’$\mathrm{g},\mathrm{g}$$\mathrm{c}(f)^{*}\subset c(g)^{*}$$\mathrm{g},\mathrm{g}$$I$$\mathrm{c}(f’)^{*}\subset c(\oint)^{*}$$J_{1}$$J_{2}$$(IJ_{1})^{*}\subset(IJ_{2})^{*}$$I$$(\mathrm{b}, \cdots, b_{m})$$J_{1}$$I$$J_{2}$$=$$\mathrm{a}\mathrm{i},$$\cdots$$J_{2}=(\mathrm{q}, \cdots, c_{l})$. Then $\mathrm{c}(fg)=IJ_{1},c(fh)=IJ_{2}$ . Since, $(fg)/(fh)=g/h$ and, we have’. That is,.
e.a.b.

$h= \sum \mathrm{q}.X^{:(n+1)}$$\mathrm{c}(fg)^{*}\subset c(fh)^{*}$$\mathrm{c}(g)^{*}\subset c(h)$In the following, let $D$ be adomain, and let$J_{1}^{*}\subset J_{2}^{*}$$A$$\mathrm{H}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}*\mathrm{i}\mathrm{s}$be aMarot ring with propert$\mathrm{y}$13Theorem 1.
astar-0peration on . The following conditions areequivalent :(i)e.a.b.
(ii) If $f/g=f’/g’$ , where $f,g,f’,g’\in A[X;S]$ with $g,g’$ regular, and if, then.
(2)astar-0petration on . The following conditions are equivalent:$A$$\mathrm{L}\mathrm{e}\mathrm{t}*\mathrm{b}\mathrm{e}$$*\mathrm{i}\mathrm{s}$$\mathrm{c}(f)^{*}\subset c(g)^{*}$$\mathrm{c}(f’)^{*}\subset c(g’)^{*}$$S$$\mathrm{L}\mathrm{e}\mathrm{t}*\mathrm{b}\mathrm{e}$(i)e.a.b.
(ii) If, where $f,g$ , $f’,g’\in D[X;S]$ with $g,g’$ non-zero, and if, then.
(3)asemistar-0peration on $A$ . The following conditions are equivalent:(i)e.a.b.
(ii) If $f/g=f’/g’$ , where $f,g,f’,g’\in A[X;S]$ with $g,g’$ regular, and if$*\mathrm{i}\mathrm{s}$$f,/.\mathit{0}=f’/\mathit{9}$$\mathrm{e}(f)^{*}\subset e(g)^{*}$$\mathrm{e}(f’)^{*}\subset e(g’)^{*}$$\mathrm{L}\mathrm{e}\mathrm{t}*\mathrm{b}\mathrm{e}$$*\mathrm{i}\mathrm{s}$$\mathrm{c}(f)^{*}\subset c(g)^{*}$(4), then.
asemistar-0peration on . The following conditions are equiv$\mathrm{c}(f’)^{*}\subset c(g’)^{*}$$S$$\mathrm{L}\mathrm{e}\mathrm{t}*\mathrm{b}\mathrm{e}$alent:(i)e.a.b.
(ii) If $f/g=f’/g’$ , where $f,g$ , $f’,d\in D[X;S]$ with $g,g’$ non-zero, and if, then.
$*\mathrm{i}\mathrm{s}$$\mathrm{e}(f)^{*}\subset e(g)^{*}$$\mathrm{e}(f’)^{*}\subset e(g’)^{*}$The proof of Theorem 1is similar to that of Proposition 1.
Let be acommutative ring. If every finitely generated regular ideal of $R$
is principal, $R$ is called an -Bezout ring. If every finitely generated regular
ideal of is invertible,
is called aPr\"ufer ring. Amultiplicative system ofconsisting of regular elements is called aregular multiplicative system of$R$ ,and aquotient ring of with respect to aregular multiplicative system iscalled aregular quotient ring of .
a star-0peration (resp. semistar-operation) on a -monoid . If theset {I is afinitely generated ffactional ideal of } is agroup under the sum, then is caud a Pr\"ufer*-multiplication semigroup.
Assume that
an e.a.b. star-0peration (resp. semistar-0peration) on ,let $D$ be adomain. Then the ring $S_{*}^{D}=\{f/g|f,g\in D[X;S]-\{0\}$ with$R$$\mathrm{r}$

$R$

$R$

$R$

$R$

$R$

$\mathrm{L}\mathrm{e}\mathrm{t}*\mathrm{b}\mathrm{e}$$\mathrm{g}$$I^{*}|$$S$$S$$(I_{1}^{*}, I_{2}^{*})\mapsto(I_{1}^{*}+I_{2}^{*})^{*}$$S$$*\mathrm{i}\mathrm{s}$

$S$

14
$e(f)^{*}\subset e(g)^{*}\}\cup\{0\}$$*\mathrm{a}\mathrm{n}\mathrm{d}D$is called the Kronecker function ring of$S$with respect to.

I isastar-0peration (resp. semistar-0peration) on $R$ . If the set {afinitely generated regular fractional ideal of $R$ } is agroup under the product, then is called a Priifer -multiplication ring. Assume
an e.a.b. star-0peration (resp. semistar-0peration) on . Then the
that
ring
}
{ $f/g|f,g\in A[X;S]-\{0\}$ with regular and.
is called the Kronecker function ring of with respect
Let $P$ be aprime ideal of $R$. The overring { $\in L|sx\in R$ for some.
$s\in R-P\}$ of $R$ is denoted by$I^{*}|$$\mathrm{L}\mathrm{e}\mathrm{t}*\mathrm{b}\mathrm{e}$$R$$(I_{1}^{*}, I_{2}^{*})-(I_{1}^{*}I_{2}^{*})^{*}$

$*$

$A$

$*\mathrm{i}\mathrm{s}$$A_{*}^{S}=$$c(f)^{*}\subset c(g)^{*}$

$g$

$A$

$\cup\{0\}$$\mathrm{t}\mathrm{o}*\mathrm{a}\mathrm{n}\mathrm{d}S$$x$$R_{[P]}$

be an e.a.b. star-0peration on , and let $T=\{g|g$ is
Theorem 2.
. Then the folowing conditions
aregular element of $A[X;S]$ withare equivalent:(0) $A[X;S]_{T}$ is aPriifer ring.
(1) is aPr\"ufer*-multiplication ring.
$A$$\mathrm{L}\mathrm{e}\mathrm{t}*$$\mathrm{c}(g)^{*}=A\}$$A$(2) $A[X;S]_{T}=A_{*}^{S}$ .
(3) $A[X;S]_{T}$ is an -Bezout ring.
(4) Each regular prime ideal of $A[X;S]_{T}$ is the extension of aprime ideal of$\mathrm{r}$

$A$

.

(5)
is aregular quotient ring of $A[X;S]$ .
(6) Each prime ideal of $A[X;S]_{T}$ is the contraction of aprime ideal of.
(7) Each regular prime ideal of $A[X;S]_{T}$ is the contraction of aprime ideal$A_{*}^{S}$$A_{*}^{S}$of$A_{*}^{S}$.

, where $P$ is
is of the form
is avaluation overring of $A$ .
aprime ideal of such that(9)
is aflat $A[X;S]$ -module.
Moreover, thre exists aPr\"ufer Marot ring $A$ with property (A) which satisfies
any e.a.b. -operation on . Then there
the following conditions: Letexists aprime ideal ofwhich is not the extension of aprime idealis the -monoid of non-negative integers.
of , where(8) Each valuation overring of$A$$A[X;S]\iota PA[\mathrm{x}_{j}s]]$$A_{*}^{S}$$A_{[P]}$$A_{*}^{S}$$*\mathrm{b}\mathrm{e}$

$A$

$*$

$A[X;\mathrm{Z}_{0}]_{T}$$A$$\mathrm{Z}_{0}$$\mathrm{g}$

For the proof of equivalence of (0) $\sim(9)$ we confer [M3, Propositions 3.1 and,,,3.9 and Theorem 3.7]. Let be afield, let
be indeter$k$$X_{1},X_{2}$$\cdots$$\mathrm{Y}_{1},\mathrm{Y}_{2}$$\cdots$

15
be aPriifer domain. Let, and let $A=R\oplus D_{0}$ , where, andis the idealofgenerated by the subset.
Then is such aring (cf.
, Theorem (1.3)]).
Asimilar result to Theorem 2holds for semistar-0perations on the ringas follows.
minates, and let$D_{0}$$R=k[[X_{1},X_{2}, \cdots,\mathrm{Y}_{1},\mathrm{Y}_{2}, \cdots]]_{1}/$$(X\dot{.}X_{j}, \mathrm{Y}_{i}\mathrm{Y}_{j_{-}}|i\neq j)$$k[[X_{1},X_{2}, \cdots,\mathrm{Y}_{1},\mathrm{Y}_{2}, \cdots]]_{1}=$$\bigcup_{n=1}^{\infty}k[[X_{1}, X_{2}, \cdots,X_{n}, \mathrm{Y}_{1}, \mathrm{Y}_{2}, \cdots, \mathrm{Y}_{n}]]$$(X_{\dot{1}}Xj,\mathrm{Y}\dot{\iota}\mathrm{Y}j|i\neq j)$$k[[X_{1},X_{2}, \cdots,\mathrm{Y}_{1}, \mathrm{Y}2, \cdots]]_{1}$$A$$\{X.\cdot X_{j},\mathrm{Y}.\cdot \mathrm{Y}_{j}|i\neq j\}$$\lfloor \mathrm{M}r1$$A$

Theorem 3.
an e.a.b. semistar-0peration on , and let $W=\dot{\{}g|g$
is aregular element ofsuch that. Then the following
conditions are equivalent:(0) $A^{*}[X;S]_{W}$ is aPriifer ring.
(1)
is aPr\"ufer*-multiplication ring.
(2)
coincides with the Kronecker function ringof with$A$$\mathrm{L}\mathrm{e}\mathrm{t}*\mathrm{b}\mathrm{e}$$A^{*}[S]$$\mathrm{c}(g)^{*}=A^{*}\}$$A$$A^{*}[X\cdot\dot, S]_{W}$$A_{*}^{S}$$A$

respect to.
(3)
is an -Bezout ring.
(4) Each regular prime ideal of $A^{*}[X;S]_{W}$ is the extension of aprime idealof .
(5)
is aregular quotient ring of $A^{*}[X;S]$ .
(6) Each prime ideal of $A^{*}[X;S]_{W}$ is the contraction of aprime ideal of.
(7) Each regular prime ideal of $A^{*}[X;S]_{W}$ is the contraction of aprime idealof.
(8) Each valuation overring of
is of the form $A^{*}[X;S]_{[QA[X_{j}S]]}*$ , where $Q$
is aprime ideal ofsuch thatis avaluation overring of .
(9)
is aflat-module.
$*\mathrm{a}\mathrm{n}\mathrm{d}S$$A^{*}[X_{\dot{J}}.S]_{W}$$\mathrm{r}$$A^{*}$$A_{*}^{S}$$A_{*}^{S}$$A_{*}^{S}$$A_{*}^{S}$$A^{*}$$A_{*}^{S}$$(A^{*})_{[Q]}$$A^{*}$$A^{*}[X_{\dot{l}}.S]$

For the proof we confer [M3, Propositions 3.2, 3.8 and 3.9].

Theorem 4. Let $D$ be adomain, and let
an e.a.b. star-0peration
on ag-monoid
and let $T=\{g|g$ is anon-zero element of $D[X;S]$ with. The following conditions are equivalent:(0) $D[X;S]_{T}$ is aPr\"ufer ring.
(1) is a Pr\"ufer*-multiplication semigroup.
(2) $D[X;S]_{T}$ coincides with the Kronecker function ing
of with respect$*\mathrm{b}\mathrm{e}$$S_{j}$$\mathrm{e}(g)^{*}=S\}$$S$$S_{*}^{D}$$\mathrm{S}$16$\mathrm{t}\mathrm{o}*\mathrm{a}\mathrm{n}\mathrm{d}D$.

(3) $D[X;.S]_{T}$ is aBezout ring.
(4) Each prime ideal of $D[X;S]_{T}$ is the extension of aprime ideal of .
(5)
is aquotient ring of $D[X;S]$ .
.
(6) Each prime ideal of $D[X;S]_{T}$ is the contraction of aprime ideal of
is of the form $D[X;S]_{PD[X;S]}$ , where $P$ is(7) Each valuation overring of
is avaluation oversemigroup of .
aprime ideal of such that(8)
is aflat $D[X;S\rfloor’$ -module.
$S$$S_{*}^{D}$$S_{*}^{D}$$S_{*}^{D}$

$S$

$S$

$S_{P}$$S_{*}^{D}$

For the proof we confer [MS, Theorems 8and 25].
Asimilar result to Theorem 4holds for semistar-0perations on .
$S$an e.a.b. semistar-0peration on , and let $W=$
Theorem 5. Let}. The following{ $g|g$ is anon-zero element of $D[X;S^{*}]$ such that:conditions are
is aPriifer ring.
(0)(1) is aPr\"ufer*-multiplication semigroup.
$S$$*\mathrm{b}\mathrm{e}$$\mathrm{e}(g)^{*}=S^{*}$$\Re \mathrm{u}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{t}$$D_{\mathrm{t}}\mathrm{f}X;S^{*}]_{W}$

$S$

.
(2)
(3) $D[X;S^{*}]_{W}$ is aBezout ring.
(4) Each prime ideal of $D[X;S^{*}]_{W}$ is the extension of aprime ideal of .
(5)
is aquotient ring of $D[X;S^{*}]$ .
.
(6) Each prime ideal of $D[X;S^{*}]_{W}$ is the contraction of aprime ideal of
is of the form $D[X;S^{*}]_{QD[X;S]}.$ , where(7) Each valuation overring ofsuch that $(S^{*})Q$ is avaluation oversemigroup of .
is aprime ideal of(8)
is aflat $D[X;S^{*}]$-module.
$D[X;S^{\mathrm{s}}]_{W}=S_{*}^{D}$$S^{*}$$S_{*}^{D}$$S_{*}^{D}$$Q$$S_{*}^{D}$$S^{*}$$S^{\mathrm{s}}$$S_{*}^{D}$

For the proof we confer [M2, Proposition 4and Theorem 23].
Appendix
Theorem. Let be a -monoid, and let be an extension semigroup. If$T$ is aNoetherian semigroup, and if
is afinitely generated -module, thenis aNoetherian semigroup.
$S$

$T$

$\mathrm{g}$

$T$

$S$

$S$

17
REFERENCES[M1] R. Matsuda, Generalizations of multiplicative ideal theory to commutive rings with zerodivisors, Bull. Fac. Sci.,Ibaralki Univ. 17(1985),49-101.
[M2] R. Matsuda, Kronecker function rings of semistar-0perations on semigroups, Math. J. Toyama Univ. 19(1996),159-170.
-operations on rings,[M3] R. Matsuda, Kronecker function rings.
Alg. Colloq.
[M4] R. Matsuda, Alocalization of asemigroup ring, Proc. 4th Sympos. on
and Computation, 2000, 57-64.
[MS] R. Matsuda and K. Sat\^o, Kronecker function rings of semigroups, Bull.
Fac. Sci.,Ibaraki Univ. 19(1987),31-46$\mathrm{o}\mathrm{f}\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{r}\sim$$\overline{\mathfrak{o}}(1998),241-254$$\mathrm{A}\mathrm{l}\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a},\mathrm{L}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{s}$