On the Central Series of a Ring

Extract

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 ⊃ … ⊃ R_α = R_α+1is the lower central series for the ring R if R_γ+1=RR_γ + R_γR for ordinal number γ and R_γ∩_δ<γ R_δ if γ is a limit ordinal. The upper central series for R is the series 0=J₀ ⊂ J₁ ⊂ … ⊂j_β = J_β+1 where J_γ+1={x ∊ R:xR + Rx ⊆ J_γ} for every ordinal number γ and J_γ = ∪ ∩δ<γ J_δ if γ is a limit ordinal. The length of the upper central series is the smallest ordinal number γ for which J_β = J_β+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.