A remark on the unique factorization theorem
A remark on the unique factorization theorem
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DOI：
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发表时间：
1957
期刊：
影响因子：
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通讯作者：
M. Nagata
中科院分区：
文献类型：
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作者：
M. Nagata
It is well known that the ring $K[x_{1}, x_{7},cdots, x_{l}]/(sum_{i=1}^{n}x_{i}^{9})$ is a unique factorization ring if $K$ is a field of characteristic different from 2 and if $ngeqq 5^{star)}$ . But it seems to the writer that the known proofs are not so simple. Theorems 1 and 2 in the present note cover the fact and our proof is simpler than the known proofs. LEMMA 1. Let $x$ be a non-zero element of a Noetherian integral domain $0$. If $xo$ is a prime ideal and if $0[1/x]$ is a unique factorization ring, then $0$ is also a unique factorization ring. PROOF. We have only to show that every prime ideal $mathfrak{p}$ of rank 1 in $0$ is principal. If $xin mathfrak{p}$ , then $P=xo$ and we assume that $x
otin mathfrak{p}$ . Let $f$ be an element of $mathfrak{p}$ such that $fo[1/x]=mathfrak{p}0[1/x]$ . Since $xfrac{rdagger^{-}}{ackslash vdash}mathfrak{p}$ , we may assume that $f
otin xo$. Let $p$ be an element of $mathfrak{p}$ . Let $r$ be the smallest integer such that $x^{r}pin fmathfrak{o}$. If $gamma$ is positive, then the element
It is well known that the ring $K[x_{1}, x_{7},cdots, x_{l}]/(sum_{i=1}^{n}x_{i}^{9})$ is a unique factorization ring if $K$ is a field of characteristic different from 2 and if $ngeqq 5^{star)}$ . But it seems to the writer that the known proofs are not so simple. Theorems 1 and 2 in the present note cover the fact and our proof is simpler than the known proofs. LEMMA 1. Let $x$ be a non-zero element of a Noetherian integral domain $0$. If $xo$ is a prime ideal and if $0[1/x]$ is a unique factorization ring, then $0$ is also a unique factorization ring. PROOF. We have only to show that every prime ideal $mathfrak{p}$ of rank 1 in $0$ is principal. If $xin mathfrak{p}$ , then $P=xo$ and we assume that $x
otin mathfrak{p}$ . Let $f$ be an element of $mathfrak{p}$ such that $fo[1/x]=mathfrak{p}0[1/x]$ . Since $xfrac{rdagger^{-}}{ackslash vdash}mathfrak{p}$ , we may assume that $f
otin xo$. Let $p$ be an element of $mathfrak{p}$ . Let $r$ be the smallest integer such that $x^{r}pin fmathfrak{o}$. If $gamma$ is positive, then the element