For an abelian p-group G,
denote the endomorphism ring of G by E(G), the ideal of small endomorphisms by
Es(G) and the quotient ring E(G)∕Es(G) by S(G). It is not difficult to show
that for a large subgroup L of G, the map that sends an endomorphism
of G to its restriction on L induces a monomorphism S(G) → S(L). We
show that if B1 is a large subgroup of a group B2 which is a direct sum
of cyclic p-groups and is of cardinality not more than 2ℵ0 and R1 and R2
are suitable subgroups of E(B1) and E(B2), then there are groups G1 and
G2 having B1 and B2 as basic subgroups such that G1 is large in G2 and
S(Gi)≅Rx∕(Es(Bi) ∩ Rt),(i = 1,2).
