Ring $R_{ 97 }$

Puninski's triangular serial ring


Let $\mathbb Z_{(2)}$ be the localization of the integers at $(2)$ ($R_{69}$), and $\mathbb Z_{2^\infty}$ be the Prüfer $2$ group, and $\mathbb Z_2=End(\mathbb Z_{2^\infty})$ ($M_{17}$) be the $2$-adic integers ($R_{84}$). Then $R$ is the ring $\begin{bmatrix}\mathbb Z_{(2)}& \mathbb Z_{2^\infty}\\0&\mathbb Z_2 \end{bmatrix}$

Notes: Puninski used any prime $p$, not just $2$. Krull dimension $1$, (probably Rentschler-Michler Krull dimension) The radical is the union of the singular ideals. It is its own classical quotient ring.

Keywords completion localization triangular ring


  • G. Puninski. Serial rings. (2001) @ Example 9.15 p 120

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  • = does not have the property
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Name Measure
cardinality $\mathfrak c$
Name Description
Right singular ideal $\begin{bmatrix}p\mathbb Z_(p)&Z_{p_\infty}\end{bmatrix}$