Have I interpreted the question correctly? If so, then I think that the result is false.

For , denote by its image in the quotient ring I assume that denotes the set of all matrices with entries in with determinant and that the map from to is the natural quotient map given by

Now take m = 7 and let Then but any lifting of A to a matrix in will have determinant of the form 8+28k (mod 49), which can never be equal to 1.

Edit.Stupid mistake: 8 + 28k can be equal to 1 (mod 49). In fact, 8 + 12×28 = 7^3 + 1; and in fact is a lifting of that matrix A to an element of So it looks to me as though the result may be true after all. But it's not at all obvious!