環理論/素理想

素理想定義:

設$R$ 為一個環。$R$ 的素理想是一個理想$p\leq R$，使得只要$I,J\leq R$ 是$R$ 的理想，且$I\cdot J\subseteq p$，則要麼$I\subseteq p$，要麼$J\subseteq p$。

素環定義:

如果一個環$R$ 的零理想是素理想，則稱該環為素環。

素理想刻畫命題:

設$R$ 為一個環，且$P\leq R$ 為一個理想。以下陳述等價：

1. $P$ 是$R$ 的素理想
2. $R/P$ 是素環
3. 無論何時$I,J\leq R$ 是$R$ 中的左理想，則$I\cdot J\subseteq P\Rightarrow (I\subseteq P\vee J\subseteq P)$
4. 無論何時$I,J\leq R$ 是$R$ 中的右理想，則$I\cdot J\subseteq P\Rightarrow (I\subseteq P\vee J\subseteq P)$

5. 無論何時 $x,y\in R$ 滿足 $xRy\in P$，則要麼 $x\in P$，要麼 $y\in P$。

Proof: We'll prove $1.\Rightarrow 2.\Rightarrow 5.\Rightarrow 3.\Rightarrow 1.$, since $3.\Leftrightarrow 4.$ follows by symmetry. Suppose first that $P$ is a prime ideal. Let ${\overline {I}},{\overline {J}}\leq R/P$ so that ${\overline {I}}\cdot {\overline {J}}\subseteq \{0+P\}$, the zero ideal of $R/P$. Then if $\pi :R\to R/P$ is the projection, consider $I:=\pi ^{-1}({\overline {I}})$, $J:=\pi ^{-1}({\overline {J}})$. Then $I\cdot J\subseteq \pi ^{-1}({\overline {I}}\times {\overline {J}})\subseteq \{0+P\}$ (since $\pi$ is a ring homomorphism), so that without loss of generality $I\subseteq P$, and hence ${\overline {I}}\leq \{0+P\}$. Suppose now that $R/P$ is a prime ring. Let then $x,y\in R$ such that $xRy\in P$. We use the bar notation ${\overline {z}}:=\pi (z)$ ($\pi :R\to R/P$ being the projection) for $z\in R$. Then we get that ${\overline {xzy}}\in R/P$ is zero for all $z\in R$. Then define the ideals $K:=\langle {\overline {x}}\rangle \leq R/P$ and $L:=\langle {\overline {y}}\rangle$, so that then $K\cdot L\leq \{0+P\}$, hence without loss of generality $K=\{0+P\}$ and hence $\pi ^{-1}(K)\leq P$ and thus $x\in P$. Suppose now that 5. holds, and let $I,J\leq R$ be left ideals such that $I\cdot J\leq P$. Suppose that there existed $x,y\in R$ so that $x\in I\setminus P$ and $y\in J\setminus P$. Then still $xRy\in P$, a contradiction. Note finally that $3.\Rightarrow 1.$ is trivial. $\Box$