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Given an integer $n$, we introduce the integral Lie ring of partitions withbounded maximal part, whose elements are in one-to-one correspondence tointeger partitions with parts in $\{1,2,\dots, n-1\}$. Starting from an abeliansubring, we recursively define a chain of idealizers and we prove that thesequence of ranks of consecutive terms in the chain is definitely periodic.Moreover, we show that its growth depends of the partial sum of the partial sumof the sequence counting the number of partitions. This work generalizes ourprevious recent work on the same topic, devoted to the modular case wherepartitions were allowed to have a bounded number of repetitions of parts in aring of coefficients of positive characteristic.

A definitely periodic chain in the integral Lie ring of partitions