The amalgamated ring R ⊲⊳^f J is a pm-ring precisely when R is a pm-ring and a specific maximal-ideal counting condition holds for each prime of S not containing J.
Azimi, Constructing (non-)Catenarian rings , J
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On the prime spectrum of the amalgamations
The amalgamated ring R ⊲⊳^f J is a pm-ring precisely when R is a pm-ring and a specific maximal-ideal counting condition holds for each prime of S not containing J.