Matsuda, Ryuki

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TitleON INERT EXTENSIONS OF GRADED INTEGRALDOMAINS(Semigroups, Formal Languages andCombinatorics on Words)Author(s)Matsuda, RyukiCitation数理解析研究所講究録 (1995), 910: 55-59Issue Date1995-05URL http://hdl.handle.net/2433/59538RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityON INERT EXTENSIONSOF GRADED INTEGRAL DOMAINS(Ry\^uki Matsuda)Faculty of Science, Ibaraki UniversityA torsion-free cancellative commutative semigroup (written additively)$\ni 0$ is called a torsionless grading monoid. By a graded integral do-main $\oplus_{S}R_{s}$ , we mean an integral domain graded by a torsionless gradingmonoid $S$ with the assumption that each $R_{s}$ is nonzero.In [1], Anderson-Anderson investigate different ways to $\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{e}\oplus sR_{s}$ .Let $\Gamma$ be a torsionless grading monoid, $S$ be a submonoid of $\Gamma$ and$\oplus \mathrm{r}R_{\alpha}$ be a $\Gamma$-graded integral domain.We define an extension of domains $D\subset E$ to be inert if whenever$0\neq xy\in D$ for some $x,$ $y\in E$ , then $xu,$ $yu^{-1}\in D$ for some unit $u$ of $E$ .We define $D\subset E$ to be strongly inert if whenever $0\neq xy\in D$ for some$x,$ $y\in E$ , then $x,$ $y\in D$ . We say that $S$ is saturated in $\Gamma$ if whenever$\alpha+\beta\in S$ for some $\alpha,$ $\beta\in\Gamma$ , then $\alpha,$ $\beta\in S$ .Among other theorems, Anderson-Anderson proved the following,Theorem $1([1,\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}3.8])$ . Let $\oplus_{\Gamma}R_{\alpha}$ be a graded integral do-main. Then $\oplus sR_{s}\subset\oplus_{\Gamma}R_{\alpha}$ is strongly inert if and only if $S\subset\Gamma$ issaturated.Anderson-Anderson state: It would be very interesting to determinenecesary and sufficient conditions on $S\mathrm{f}\mathrm{o}\mathrm{r}\oplus sR_{s}\subset\oplus \mathrm{r}R_{\alpha}$ to be an inertextension.The aim of this paper is to determine necessary and sufficient con-ditions on $S$ for $\oplus_{S}R_{s}\subset\oplus_{\Gamma}R_{\alpha}$ to be an inert extension.The quotient group $\{s_{1}-S_{2}|s_{i}\in S\}$ of $S$ is denoted by $\mathrm{q}(S)$ . $\mathrm{q}(S)$ isa subgroup of $\mathrm{q}(\Gamma)$ and $\mathrm{q}(\Gamma)$ is a totally ordered abelian group.Lemma 2. Assume that $\oplus_{S}R_{s}\subset\oplus_{\Gamma}R_{\alpha}$ is inert. Set $\Gamma\cap \mathrm{q}(S)=T$ .910 1995 55-59 55Then $\oplus_{S}R_{s}\subset\oplus_{T}R_{t}$ is inert.Proof Assume that $0\neq\varphi_{1}\varphi_{2}\in\oplus_{S}R_{s}$ for $\varphi_{1},$ $\varphi_{2}\in\oplus_{T}R_{t}$ . Then$\varphi_{1}u,$$\varphi_{2}u-1\in\oplus sR_{s}$ for some unit $u\in R_{\alpha_{1}}$ of $\oplus \mathrm{r}R_{\alpha}$ . It follows that$\alpha_{1},$$-\alpha_{2}\in^{\tau}$ .The submonoid { $\alpha\in\Gamma|\alpha+\alpha’\in S$ for some $\alpha’\in\Gamma$ } of $\Gamma$ is calledsaturation of $S$ in $\Gamma$ .Lemma 3. Assume that $\oplus_{S}R_{s}\subset\oplus \mathrm{r}R_{\alpha}$ is inert. Set $\Gamma\cap \mathrm{q}(S)=T$and let $\Gamma^{*}$ be the saturation of $S$ in $\Gamma$ . Then $T\subset\Gamma^{*},$ and $\oplus_{T}R_{t}\subset\oplus \mathrm{r}*R_{\beta}$is inert.Proof. Assume that $0\neq F_{1}F_{2}=f\in\oplus\tau R_{t}$ for some $F_{1},$ $F_{2}\in\oplus_{\Gamma^{*}}R\beta$and $f\in\oplus_{T}R_{t}$ . Then we have $fa\in\oplus_{S}R_{s}$ for some $0\neq a\in R_{s_{1}}$ and$s_{1}\in S$ . Hence we have $F_{1}u,$ $F_{2}au^{-}1\in\oplus sR_{s}$ for some unit $u\in R_{\alpha_{1}}$ of$\oplus_{\Gamma}R_{\alpha}$ . Then we have $\alpha_{1},$ $-\alpha_{1}\in\Gamma^{*}$ . Therefore $F_{2}u^{-1}\in\oplus_{T}R_{t}$ .Lemma 4. Let $\Gamma^{*}$ be the saturation of $S$ in $\Gamma$ . $\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{n}\oplus \mathrm{r}*R_{\beta}\subset\oplus_{\Gamma}R_{\alpha}$is strongly inert.Proof. By Theorem 1.Let $x\in\oplus_{S}R_{s}$ , with $x=x_{1}+\cdots+x_{n}$ , where $0\neq x_{i}\in R_{s:}$ for each $i$ and$s_{1}<\cdots<s_{n}$ . Then the subset $\{s_{1}, \cdot\cdot, , s_{n}\}$ of $S$ is called support of $x$ ,and is denoted by $\mathrm{s}_{\mathrm{u}\mathrm{p}\mathrm{p}}(X)$ . Let $I_{1},$ $I_{2}$ be subsets of $\mathrm{q}(S)$ . Then the subset{ $x_{1}+x_{2}|x_{i}\in I_{i}$ for each $i$ } of $\mathrm{q}(S)$ is denoted by $I_{1}+I_{2}$ . $\sim I+\cdots+I$ is$n$denoted by $nI$ for $I\subset \mathrm{q}(S)$ . The subset $\{\alpha\in I|I_{1}+\alpha\subset I_{2}\}$ is denotedby $(I_{2} : I_{1})_{I}$ . Next, the subset { $s\in S|R_{s}$ contains a unit $\mathrm{o}\mathrm{f}\oplus sR_{s}$ } isdenoted by $S^{(0)}$ .Lemma 5 Assume that $\oplus_{S}R_{s}\subset\oplus_{\Gamma}R_{\alpha}$ is inert. Let $I_{1},$ $I_{2}$ benon-empty finite subsets of $\Gamma$ such that $I_{1}+I_{2}\subset S$ . Then we have$(S : I_{1})\mathrm{r}(0)+(s : I_{2})\Gamma(0)\ni 0$.56Proof. We take $F_{1},$ $F_{2}\in\oplus_{\Gamma}R_{\alpha}$ such that $\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(F1)=I_{1},$ $\mathrm{S}\mathrm{u}_{\mathrm{P}\mathrm{p}}(p_{2})=$$I_{2}$ . Then $F_{1}F_{2}\in\oplus sR_{s}$ . Hence we have $F_{1}u,$ $F_{2}u^{-1}\in\oplus sR_{s}$ for some unit$u\in R_{\alpha_{1}}$ of $\oplus_{\Gamma}R_{\alpha}$ . Then $\alpha_{1}\in(S:I_{1})_{\Gamma}(0)$ and $-\alpha_{1}\in(S:I_{2})\Gamma(0)$ .Lemma 6. Assume that $\oplus_{S}R_{s}\subset\oplus_{\Gamma}R_{\alpha}$ is inert. Then $S$ is inte-grally closed in $\Gamma$ .Proof. Assume that $0\neq\alpha_{1}\in\Gamma$ is integral over $S$ . We have $n\alpha_{1}\in S$for some $n\in$ N. Take $0\neq x\in R_{\alpha_{1}}$ . We have $\oplus_{S}R_{s}\ni 1-x^{n}=$$(1-x)(1+x+\cdots+x^{n-1})$ . Hence there exists aunit $u\in R_{\alpha_{2}}\mathrm{o}\mathrm{f}\oplus \mathrm{r}R_{\alpha}$such that $(1-x)u,$ $(1+x+\cdots+X^{n-})1u^{-1}\in\oplus_{S}R_{s}$ . Then it $\mathrm{f}\mathrm{o}\mathrm{U}_{0}\mathrm{w}\mathrm{s}\alpha 1\in S$.We say that $S$ satisfies $(^{*})$ in $\Gamma$ if whenever $n\alpha\in \mathrm{q}(S)$ for some $\alpha\in\Gamma$and $n\in \mathrm{N}$ , then $\alpha\in S$ .Lemma 7. Assume that $\oplus_{S}R_{s}\subset\oplus_{\Gamma}R_{\alpha}$ is inert. Set $\Gamma\cap \mathrm{q}(S)=T$and let $\Gamma^{*}$ be the saturation of $S$ in $\Gamma$ . Then $T$ satisfies $(^{*})$ in $\Gamma^{*}$ .Proof. Assume that $n\beta\in \mathrm{q}(T)$ for some $\beta\in\Gamma^{*}$ and $n\in$ N. Since$\mathrm{q}(T)=\mathrm{q}(S)$ , we have $n\beta+s\in S$ for some $s\in S$ . Then $n(\beta+s)\in S$ .Lemma 6implies that $\beta+s\in S$ . Hence $\beta\in T$ .Let $\alpha_{1},$ $\alpha_{2}\in\Gamma$ . If $n(\alpha_{1}-\alpha_{2})\in \mathrm{q}(S)$ for some $n\in \mathrm{N}$ , we define$\alpha_{1}\sim_{\mathrm{t}}s,\mathrm{r})\alpha_{2}$ . Then $\sim_{\mathrm{t}}s,\mathrm{r}$) is an equivalence relation on $\Gamma$ . The equiva-lence class of $\alpha\in\Gamma$ is denoted by $\overline{\alpha}$ . For $\alpha\in\Gamma$ , we define $R_{\overline{\alpha}}=\Sigma_{\alpha\in\overline{\alpha}\alpha},R’$ .Lemma 8 $([1,\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}3.1])$. Assume that $S$ satisfies $(^{*})$ in F. Thenthe quotient set $\overline{\Gamma}$ of $\Gamma \mathrm{b}\mathrm{y}\sim_{\mathrm{t}}s,\mathrm{r}$) is a grading monoid. Moreover, $\oplus_{\Gamma}R_{\alpha}=$$\oplus_{\overline{\Gamma}}R_{\overline{\alpha}}$ is a $\overline{\Gamma}$-graded domain and $R_{\overline{0}}=\oplus_{S}R_{s}$ .Lemma 9 ([1, Proposition $3.3,(1)]$ ). $R_{0}\subset\oplus sR_{s}$ is inert if and only if$S^{\langle 0)}$ is the units of $S$ .Lemma 10. Assume that $\oplus_{S}R_{s}\subset\oplus_{\Gamma}R_{\alpha}$ is inert. Assume that $S$satisfies $(^{*})$ in $\Gamma$ and the saturation of $S$ in $\Gamma$ is $\Gamma$ . Let $\beta\in\Gamma$ . Then there57exists $\beta_{1}\in\Gamma^{(0)}$ such that $\beta\sim_{\mathrm{t}^{S,\mathrm{r}})}\beta 1$ .Proof. Then the quotient set $\overline{\Gamma}$ is a group. $\oplus \mathrm{r}R_{\alpha}=\oplus_{\overline{\Gamma}}R_{\overline{\alpha}}$ is a $\overline{\Gamma}-$graded domain and $R_{\overline{0}}=\oplus_{S}R_{s}$ by Lemma 8. By Lenuna 9, $R_{\overline{\beta}}$ containsa unit $u$ of $\oplus_{\overline{\Gamma}}R_{\overline{\alpha}}$ . Hence there exists $\beta_{1}\in\Gamma^{(0)}$ such that $\beta\sim_{\mathrm{t}s,\mathrm{r})}\beta 1$ and$u\in R_{\beta_{1}}$ .Lemma 11. Assume that $\oplus_{S}R_{s}\subset\oplus \mathrm{r}R_{\alpha}$ is inert. Let $\Gamma^{*}$ be thesaturation of $S$ in $\Gamma$ . Let $\beta\in\Gamma^{*}$ . Then there exists $\beta_{1}\in(\Gamma^{*})^{\mathrm{t})}0$ such that$\beta\sim_{\mathrm{t}}s,\mathrm{r})\beta 1$ .Proof. We set $\Gamma\cap \mathrm{q}(S)=T$ . $T$ satisfies $(^{*})$ in $\Gamma^{*}$ by Lemma 7. Lemma3 implies that $\oplus\tau R_{t}\subset\oplus \mathrm{r}*R_{\beta}$ is inert. Hence there exists $\beta_{1}\in(\Gamma^{*})^{(0)}$such that $\beta\sim(\tau,\mathrm{r}^{\mathrm{s}})\beta_{1}$ . Then we have $\beta\sim(s,\mathrm{r})\beta 1$ .Lamma 12. Assume that $S$ satisfies $(^{*})$ in $\Gamma$ and the saturationof $S$ in $\Gamma$ is F. Assume that for each $\alpha\in\Gamma$ there exists $\alpha’\in\Gamma^{(0)}$ suchthat $\alpha\sim(S,\Gamma)\alpha J.$ Then $\oplus_{S}R_{s}\subset\oplus_{\Gamma}\Gamma_{\alpha}$ is inert.Proof. The quotient set $\overline{\Gamma}$ is a group. $\oplus_{\Gamma}R_{\overline{\alpha}}=\oplus_{\overline{\Gamma}}R_{\overline{\alpha}}$ is a $\overline{\Gamma}$-gradeddomain, and $R_{\overline{0}}=\oplus_{S}R_{s}$ . Lemma 9 implies that $\oplus_{S}R_{s}\subset\oplus_{\Gamma}R_{\alpha}$ is inert.Lemma 13. Let $f,g$ be non-zero elements $\mathrm{o}\mathrm{f}\oplus s^{R}S$ and set $|\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{P}(g)|=$$k$ . Then we have $k\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{P}(f)+\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(g)=(k-1)\mathrm{S}\mathrm{u}_{\mathrm{P}}\mathrm{p}(f)+\mathrm{S}\mathrm{u}_{\mathrm{P}\mathrm{P}}(fg)$.Proof. The proof is similar with that of $[2, 6.2.\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}]$ .Lemma 14. Assume that $\Gamma\subset \mathrm{q}(S)$ and $S$ is integrally closed in$\Gamma$ . Assume that $(S : I_{1})_{\Gamma^{(0})}+(S : I_{2})_{\Gamma^{()}}0\ni 0$ for every non-empty finitesubsets $I_{1},$ $I_{2}$ of $\Gamma$ such that $I_{1}+I_{2}\subset S.$ Then $\oplus_{S}R_{s}\subset\oplus_{\Gamma}R_{\alpha}$ is inert.Proof Assume that $0\neq F_{1}F_{2}\in\oplus sR_{s}$ for some $F_{1},$ $F_{2}\in\oplus \mathrm{r}R_{\alpha}$ .Lemma 13 implies that $(m+1)\mathrm{s}\mathrm{u}_{\mathrm{P}\mathrm{p}(}F1)+\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(F2)=m\mathrm{S}\mathrm{u}_{\mathrm{P}\mathrm{p}(}F1)+\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}$$(F_{1}F_{2})$ for some $m\in \mathrm{N}$ . Let $\{V_{\lambda}|\lambda\in\Lambda\}$ be the set of valuation over-semigroups of $S$ . Then $\bigcap_{\lambda}V_{\lambda}$ is the integral closure of $S$ . We have58$(m+1)\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(F1)+\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(F2)+V_{\lambda}=m\mathrm{s}\mathrm{u}_{\mathrm{P}\mathrm{P}}(F1)+\mathrm{s}_{\mathrm{u}_{\mathrm{P}\mathrm{p}}}(F1F2)+V_{\lambda}$foreach $\lambda$ . It follows that $\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{P}(F1)+^{\mathrm{s}()}\mathrm{u}\mathrm{p}\mathrm{P}F_{2}+V_{\lambda}=\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(F1F2)+V_{\lambda}\subset V_{\lambda}$for each $\lambda$ . Hence $\mathrm{S}\mathrm{u}_{\mathrm{P}\mathrm{P}}(F1)+^{\mathrm{s}_{\mathrm{u}\mathrm{p}}()}\mathrm{p}F_{2}$ is contain$‘ \mathrm{e}\mathrm{d}$ in the integral closureof $S$ . Then $\mathrm{S}\mathrm{u}_{\mathrm{P}\mathrm{P}}(F1)+^{\mathrm{s}_{\mathrm{u}\mathrm{p}}()}\mathrm{p}F_{2}\subset S$. By the assumption, there exlsts$\alpha_{1}\in(S:\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(F1))\Gamma(\mathit{0})$ with $-\alpha_{1}\in(S :\mathrm{S}\mathrm{u}_{\mathrm{P}\mathrm{P}}(F2))_{\Gamma}(0)$ . We take a unit$u\in R_{\alpha_{1}}$ of $\oplus_{\Gamma}R_{\alpha}$ . Then $F_{1}u_{1,2}Fu^{-1}\in\oplus sR_{s}$ .Theorem 15. Set $\Gamma\cap \mathrm{q}(S)=T$ and let $\Gamma^{*}$ be the saturation of $S$in $\Gamma.$ Then $\oplus_{S}R_{s}\subset\oplus^{:}\mathrm{r}R_{\alpha}$ is inert if and only if the following (1) $\sim(4)$hold.(1) $S$ is integrally closed in $\Gamma$ .(2) $(S : I_{1})_{T^{(}}0)$ $+(S : I_{2})_{T^{()}}0\ni 0$ for every non-empty finite subsets$I_{1},$ $I_{2}$ of $T$ such that $I_{1}+I_{2}\subset S$ .(3) $T$ satisfies $(^{*})$ in $\Gamma^{*}$ .(4) For each $\beta\in\Gamma^{*}$ , there exists $\beta_{1}\in(\Gamma^{*})^{(0)}$ with $\beta\sim(s,\Gamma)\beta_{1}$ .Proof. Assume that $\oplus_{S}R_{s}\subset\oplus_{\Gamma}R_{\alpha}$ is inert. Then Lemma 6 implies(1). $\oplus_{S}R_{s}\subset\oplus_{T}R_{t}$ is inert by Lemma 2. Lemma 5 implies (2). Lemma7 implies (3). Lemma 11 implies (4). Conversely, assume (1),(2) $,(3)$ and(4). Then $\oplus_{S}R_{s}\subset\oplus\tau R_{t}$ is inert by Lemma 14. For each $\beta\in\Gamma^{*}$ , thereexists $\beta_{1}\in(\Gamma^{*})^{(0)}$ with $\beta\sim(S,\Gamma)\beta_{1}$ . Then we have $\beta\sim(T,\Gamma)\beta_{1}$ . Then$\oplus_{T}R_{t}\subset\oplus \mathrm{r}*R_{\beta}$ is inert by Lemma 12. $\oplus \mathrm{r}_{*}R_{\beta}\subset\oplus_{\Gamma}R_{\alpha}$ is strongly inertby Lemma 4. It follows that $\oplus_{S}R_{s}\subset\oplus_{\Gamma}R_{\alpha}$ is inert.Remark 16. Assume that $\oplus_{\Gamma}R_{\alpha}$ is the semigroup ring $D[X;\mathrm{r}]$ overa domain $D$ with $R_{\alpha}=DX^{\alpha}$ for each $\alpha$ . Then $\Gamma^{(0)}$ is the set of units of$\Gamma$ . The conditions for $D[X;s]\subset D[X;\Gamma]$ to be inert in Theorem 15 areconditions mearly for $S\subset\Gamma$ .REFERENCES[1] $\mathrm{D}.\mathrm{D}$ .Anderson and $\mathrm{D}.\mathrm{F}$ .Anderson, Grading integral domains, Comm.Algebra 11 (1983),1-19.[2] R.Gilmer and T.Parker, Divisibility properties in semigroup rings,Michigan Math. Journ. 21(1974),65-86.59

ON INERT EXTENSIONS OF GRADED INTEGRAL DOMAINS(Semigroups, Formal Languages and Combinatorics on Words)

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ON INERT EXTENSIONS OF GRADED INTEGRAL DOMAINS(Semigroups, Formal Languages and Combinatorics on Words)

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