Let $A$ denote a unital commutative ring. Given a system of ideals $(I_p)$ indexed by a partially ordered set $P$ such that if p <= q, then $I_p$ is contained in $I_q$, when is   

    $GL_n(projlim(A/{I_p})) \cong projlim( GL_n(A/{I_p}) ) ?  

Of course this is true when $A$ is $I$-adically complete. 

What happens when $A$ is a topological ring and the $GL_n$ 's are endowed with "suitable" topologies making them toplogical groups?