If p is a prime, we call an element x ≠ 1 of a group G a generalized p-element if, for every n ≧ 1, there exists r ≧ 0 such that xpr
∈ Gn, where Gn
is the nth term of the lower central series of G. Bovdi [1] proved that if
G is a finitely generated group having a generalized p-element, and if ∩
n
Δ
n
(Z(G) = 0 where Δ(Z(G)) is the augmentation ideal, then G is residually a finite p-group.
We recall that if R is a ring, then the nth dimension subgroup of G over R, denoted by Dn(R(G)), is defined to be {g | g – 1 ∈ Δ
n
(R(G))}. In this note, we show that if G is finitely generated, then ∩
n
Dn
(Z
p
∧(G)) = 1 ⇔ ∩
n
Δ
n
(Zp
∧ (G)) = 0 ⇔ G is residually a finite p-group.