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YCor
YCor
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Let $R$ be a commutative ring with $1$ such that its uniform dimension is infinity, equivalently, $Sup\{k \mid R$ contains a direct sum of $k$ nonzero ideals$\}=\infty. $$$\sup\{k \mid R \text{ contains a direct sum of $k$ nonzero ideals }\}=\infty.$$ How can we construct an ideal $I$ of $R$ and an element $r\in R\setminus I$ such that the ring $(R/I)_{r+I}$ has infinitely many idempotents, where $(R/I)_{r+I}$ is the localization of the ring $R/I$ at the subset $\{r^n+I\mid n\in \mathbb{N}\}$ of $R/I$?

Let $R$ be a commutative ring with $1$ such that its uniform dimension is infinity, equivalently, $Sup\{k \mid R$ contains a direct sum of $k$ nonzero ideals$\}=\infty. $ How can we construct an ideal $I$ of $R$ and an element $r\in R\setminus I$ such that the ring $(R/I)_{r+I}$ has infinitely many idempotents, where $(R/I)_{r+I}$ is the localization of the ring $R/I$ at the subset $\{r^n+I\mid n\in \mathbb{N}\}$ of $R/I$?

Let $R$ be a commutative ring with $1$ such that its uniform dimension is infinity, equivalently, $$\sup\{k \mid R \text{ contains a direct sum of $k$ nonzero ideals }\}=\infty.$$ How can we construct an ideal $I$ of $R$ and an element $r\in R\setminus I$ such that the ring $(R/I)_{r+I}$ has infinitely many idempotents, where $(R/I)_{r+I}$ is the localization of the ring $R/I$ at the subset $\{r^n+I\mid n\in \mathbb{N}\}$ of $R/I$?

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Sara W.W
Sara W.W
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Infinite uniform dimension $\Rightarrow$ infinitely many idempotents in a localization of a quotient

Let $R$ be a commutative ring with $1$ such that its uniform dimension is infinity, equivalently, $Sup\{k \mid R$ contains a direct sum of $k$ nonzero ideals$\}=\infty. $ How can we construct an ideal $I$ of $R$ and an element $r\in R\setminus I$ such that the ring $(R/I)_{r+I}$ has infinitely many idempotents, where $(R/I)_{r+I}$ is the localization of the ring $R/I$ at the subset $\{r^n+I\mid n\in \mathbb{N}\}$ of $R/I$?