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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.

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  • $\begingroup$ The definition does not use commutativity, but most properties do. $\endgroup$ – darij grinberg Jun 1 at 5:54
  • $\begingroup$ Not "the determinant over noncommutative ring is not defined" but rather "the usual definition is not suitable" There are certainly definitions of determinants used with noncommutative fields. $\endgroup$ – rschwieb Jun 3 at 13:09
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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 $D$.

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