Geodesic
On the Central Series of a Ring
Canadian mathematical bulletin, Tome 16 (1973) no. 1, pp. 27-32

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The study of group types was completed by Meldrum [1]. The concept of ring type described here is based on analogous definitions.The series R = R0 ⊃ R1 ⊃ ... ⊃ Rα = Rα+1 is the lower central series for the ring R if Rγ+1=RRγ + RγR for ordinal number γ and Rγ∩δ<γ Rδ if γ is a limit ordinal. The upper central series for R is the series 0=J0 ⊂ J1 ⊂ ... ⊂jβ = Jβ+1 where Jγ+1={x ∊ R:xR + Rx ⊆ Jγ} for every ordinal number γ and Jγ = ∪ ∩δ<γ Jδ if γ is a limit ordinal. The length of the upper central series is the smallest ordinal number γ for which Jβ = Jβ+1 .The length of the lower central series is defined similarily. We shall say the ring has type (β,α) if the length of the upper central series is β and the length of the lower central series is α. 
Biggs, R. G. On the Central Series of a Ring. Canadian mathematical bulletin, Tome 16 (1973) no. 1, pp. 27-32. doi: 10.4153/CMB-1973-007-2
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