COMPLEX ALGEBRAS 
127 
§ 79] 
factors in C. Hence /(co) is of degree 1 and x is the 
product of the modulus by a complex number. 
Every complex simple algebra, not a zero algebra of 
order 1, is a simple matric algebra. For, by § 51, it is 
the direct product of a division algebra (here of order 1) 
by a simple matric algebra. 
A complex semi-simple algebra which is not simple 
is a direct sum of simple matric algebras (§ 40). 
The characteristic and rank equations of any semi 
simple complex algebra are known by §§ 71, 72. 
We are now in a position to give an elementary proof 
of the principal theorem that every complex algebra with 
a modulus is either semi-simple or is the sum of its maxi 
mal nilpotent invariant sub-algebra and a semi-simple 
sub-algebra. In the proof in § 78 of a more general 
theorem, use was made of the theorem in § 77 which 
may be proved far more simply for a complex algebra A. 
We may assume that the order of A is r> 1. Then A 
is not simple since a simple matric algebra of order 
r> 1 contains idempotent elements eu other than its 
modulus hen. In a semi-simple algebra which is not 
simple, the modulus of each component simple algebra 
is idempotent. Since A is .not semi-simple, it has a 
maximal nilpotent invariant sub-algebra N. But A — N 
is a complex division algebra (middle of § 77), which is 
therefore of order 1. Thus N is of order r — 1. Hence 
A is the sum of N and the division algebra generated by 
the modulus of A. 
For normalized basal units of any complex algebra, 
see chapter x.