  Given a graded associative algebra $A$, its lower central series is defined
by $L_1 = A$ and $L_{i+1} = [L_i, A]$. We consider successive quotients $N_i(A)
= M_i(A) / M_{i+1}(A)$, where $M_i(A) = AL_i(A) A$. These quotients are direct
sums of graded components. Our purpose is to describe the $\mathbb{Z}$-module
structure of the components; i.e., their free and torsion parts. Following
computer exploration using {\it MAGMA}, two main cases are studied. The first
considers $A = A_n / (f_1,\dots, f_m)$, with $A_n$ the free algebra on $n$
generators $\{x_1, \ldots, x_n\}$ over a field of characteristic $p$. The
relations $f_i$ are noncommutative polynomials in $x_j^{p^{n_j}},$ for some
integers $n_j$. For primes $p > 2$, we prove that $p^{\sum n_j} \mid
\text{dim}(N_i(A))$. Moreover, we determine polynomials dividing the Hilbert
series of each $N_i(A)$. The second concerns $A = \mathbb{Z} \langle x_1, x_2,
\rangle / (x_1^m, x_2^n)$. For $i = 2,3$, the bigraded structure of $N_i(A_2)$
is completely described.
