1)This book "introduction to Rings and Modules" by C. Mussili, says determinant of a matrix over non commutative ring is not defined. But in defining the determinant of a matrix where the commutativity of the ring is used.
2) Please suggest some texts to study group of units of matrix rings over non commutative rings.
You might be interested in this article:
Gelʹfand, I. M.; Retakh, V. S. Determinants of matrices over noncommutative rings. (Russian) ; translated from Funktsional. Anal. i Prilozhen. 25 (1991), no. 2, 13--25, 96 Funct. Anal. Appl. 25 (1991), no. 2, 91--102
and in this book
Okniński, Jan. Semigroups of matrices, Series in Algebra, 6. World Scientific Publishing Co., Inc., River Edge, NJ, 1998. xiv+311 pp. ISBN: 981-02-3445-7
In particular Chapter 9 is about matrix algebras $M_n(D)$ over a division algebra $D$.