Subgroups of the elementary and Steinberg groups of congruence level \(I^ 2\) (Q801030)

Let I be an ideal in the ring B. It is well known that the relative group GL(B,I) depends only on I, but that E(B,I) varies with the choice of ambient ring B. Let \(E(I)=E({\mathbb{Z}}\oplus I,I).\) The authors prove (Theorem 1) that the following subgroups of GL(I) are equal, \([E(I),E(I)]=[E(B,I),E(B,I)]=[GL(B,I),GL(B,I)]=E(I)\cap GL(I^ 2).\) However, in general \(E(I)\cap GL(I^ 2)\neq E(B,I)\cap GL(I^ 2).\) (In particular, the commutator subgroup of E(B,I) depends only on I.) They also prove similar results for subgroups of the relative Steinberg group St(B,I).