EQUIVALENT MATRICES 
§ 9 8 ] 
169 
with attention to the order of multiplication. Hence 
each d'ij is in S. 
The constant term of the rank equation of the 
general element x of D is called the norm of x and denoted 
by N{x). It is a divisor of the first determinant A(x) 
of x (§ 69). If x^o, x has an inverse y in D by the 
definition of D. By § 58, A(x)A(y) = 1, whence A(x) 5^0, 
N{x)t^o. Hence N{x) = o implies x = o. 
We shall restrict our attention to maximal sets S 
of elements of rational division algebras D which possess 
the following further property: 
P. If a and b(b^6) are any two elements of S, 
there exist elements q, c, Q, C oi S such that 
a=qb-\-c. a = bQ-\-C, 
where the norms of the remainders c and C are numeri 
cally less than the norm of the divisor b. 
Evidently property P holds for the important case 
in which D is of order 1 when the elements of D may be 
taken to be the rational numbers, so that the ele 
ments of S are integers, each being its own norm. 
Then the following investigation becomes a study of 
the arithmetic of matrices whose elements are all 
integers. 
Property P was seen in Lemma 2 of § 91 to hold 
when D is the algebra of rational quaternions. It will 
be seen in § 105 to hold also for two division algebras 
which are direct generalizations of the algebra of qua 
ternions. 
Two matrices d and d' with elements in S shall be 
called equivalent if and only if d' =pdq, where p and q