Theorem, The rank equation of the algebra of all 
n-rowed square matrices (X{j) with elements in any infinite 
field is its characteristic equation (31). 
Hence by § 70 the characteristic determinant of x is 
the nth power of R{u>) apart from sign. 
72. Rank equation of a direct sum. If an associative 
algebra A with the modulus* e over an infinitef field F is a 
direct sum of algebras A I} .... f A h and if R{co)=o is 
the rank equation of A, and Rfo) =0 is that of A i} then 
R^^Rfof) .... R t (co). 
The co-ordinates (j = 1, . . . . , nf) of the general 
element x t - of A { are independent variables in F. The 
general element x = Sx,- of A has as co-ordinates the 
independent variables fa {j=i, . . . . , m; i= 1, 
. . . . , t) in F. If also y = Hyi, then xy = 2x t -y;, whence 
x k = *, o=R{x) = '2 l R{x i ). 
Hence each R{x l )=o. By the lemma and the footnote 
in § 69, R{w) is divisible by the Rfoo) and hence by their 
least common multiple L(co) when the £# are indetermi- 
nates. Write L(oo) =Ri{u)Qi(co). ThenZ,^-) =o, whence 
L(x) =2L(xi) =0, so that L{of) is divisible by R{cf) 
by the same lemma. The two results show that R(co) 
is the least common multiple of the Rfco), 
The theorem will therefore follow if we prove that no 
two of the Rf co) have a common divisor of degree >0. 
Suppose that Rfu>) and R 2 {u>) have a common divisor 
D{co) of degree >0. Since Rf o>) is of degree o in the 
* The theorem may fail if there is no modulus since the rank equation 
of a zero algebra is always w 2 =o. 
fThe theorem fails for the algebra (ui) ® (u 2 ) ® (u 3 ), u\—Ui over 
the field of order 2, since its rank equation is linear (end of § 69), while 
that of {ui) is co—£¿=0.