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Warner
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A commutative ring $R$ with $1$ is said to be arithmetical if any of the following equivalent conditions holds: forfor all ideals $a , b$ of $R$, $ a ∩ ( b + c ) = ( a ∩ b ) + ( a ∩ c )$ or the localization $R_m$ is a uniserial ring for every maximal ideal $m$ of $R$. Now let $x\in R$, where $R$ is an arithmetical ring. Is there condition under which $R/ann_R(x)$$\frac{R}{ann_R(x)}$ is a local ring, where $ann_R(x)=\{r\in R: rx=0\}$?

A commutative ring $R$ with $1$ is said to be arithmetical if any of the following equivalent conditions holds: for all ideals $a , b$ of $R$, $ a ∩ ( b + c ) = ( a ∩ b ) + ( a ∩ c )$ or the localization $R_m$ is a uniserial ring for every maximal ideal $m$ of $R$. Now let $x\in R$. Is there condition under which $R/ann_R(x)$ is a local ring?

A commutative ring $R$ with $1$ is said to be arithmetical if for all ideals $a , b$ of $R$, $ a ∩ ( b + c ) = ( a ∩ b ) + ( a ∩ c )$ or the localization $R_m$ is a uniserial ring for every maximal ideal $m$ of $R$. Now let $x\in R$, where $R$ is an arithmetical ring. Is there condition under which $\frac{R}{ann_R(x)}$ is a local ring, where $ann_R(x)=\{r\in R: rx=0\}$?

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Warner
Warner
  • 21
  • 2

Local quotient of arithmetical ring

A commutative ring $R$ with $1$ is said to be arithmetical if any of the following equivalent conditions holds: for all ideals $a , b$ of $R$, $ a ∩ ( b + c ) = ( a ∩ b ) + ( a ∩ c )$ or the localization $R_m$ is a uniserial ring for every maximal ideal $m$ of $R$. Now let $x\in R$. Is there condition under which $R/ann_R(x)$ is a local ring?