# Ring detail

## Name: Puninski's triangular serial ring

Description: Let $\mathbb Z_{(p)}$ be the localization of the integers at a prime $p$, and $\mathbb Z_{p^\infty}$ be the Prüfer $p$ group, and $\mathbb Z_p=End(\mathbb Z_{p^\infty})$ be the $p$-adic integers. Then $R$ is the ring $\begin{bmatrix}\mathbb Z_{(p)}& \mathbb Z_{p^\infty}\\0&\mathbb Z_p \end{bmatrix}$

Notes: Krull dimension $1$, (probably Rentschler-MIchler Krull dimension) The radical is the union of the singular ideals. It is its own classical quotient ring.

Reference(s):

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

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