i74 
ARITHMETIC OF AN ALGEBRA • [chap, x 
algebra D for which properties R, C, U, P hold, is equiva 
lent to a diagonal matrix (d ly ...., d r , o, ...., 6), 
where each di is both a right and a left divisor of di+t, 
di+ 2 , Here di may be replaced by udiv, where u 
and v are any units of S. 
The final remark follows from (iv) and (v). 
We shall call (u x , . , u n ) a unit if u x , . . . . , u n 
are any units of S. Employing only matrices whose 
elements are in S, we shall call a matrix d a prime matrix 
if it is not a unit and if it admits only such representations 
as a product of two matrices in which one of them is 
a unit. 
By definition any matrix equivalent to d is of the 
form pdq where the matrices p and q are units of the 
algebra. In other words any matrix d is associated 
(§ 88) with a diagonal matrix. 
First, let S be the set of integers so that the elements 
of our matrices are integers. Then any matrix d of 
rank n will be expressible as a product of prime matrices 
in one and only one way apart from unit factors if the 
like property is proved for diagonal matrices. The 
latter is proved essentially* as at the end of § 93. Hence 
unique factorization into prime matrices holds. 
Second, let S be the set of integral quaternions. 
The uniqueness of factorization of diagonal matrices 
and hence of any matrices whose elements are integral 
quaternions is subject to the same limitations as in 
Theorem 5 of § 91. 
* We now need consider (a, ft 75) only when a divides /3 and when 
a and /3 both divide 78. For example, if 7 = /3, we employ 
(a, ft /36) = (a, ft /3) (1, 1, 5). 
While we there employed (a, /3, 1), we would now use the equivalent 
matrix (1, a, /3).