Let G be a p-primary Abelian
group. The final rank of G can be obtained in two equivalent ways: either as
infn∈ω{r(pnG)} where !r(pnG) is the rank of pnG; or as sup is a basic
subgroup of . In fact it is known that there exists a basic subgroup of G such that
r(G∕B) is equal to the final rank of G. In this paper are displayed two appropriate
generalizations of the above definitions of final rank, rα(G) and sα(G), where α is a
limit ordinal. It is shown that the two cardinals rα(G) and sα(G) are indeed the same
for any limit ordinal α. In this context one can think of the usual final rank as ω-final
rank”.
is a basic
subgroup of 
. In fact it is known that there exists a basic subgroup of 