i8o 
ARITHMETIC OF AN ALGEBRA [chap, x 
provided a p = v£i~ 1 . Multiply by v. Hence zv = x. This 
proves that, if A(z) so that each £5^0, z is associated 
with its abridgment x. Recalling the definition of 
associated arithmetics (§ 88), we have 
Theorem 2. If the y’s are rational for the algebra 
A =S+N in Theorem 1, the arithmetic of A is associated 
with the arithmetic of the sub-algebra S having the basal 
units e x , .... , eu. 
103. General complex algebra. Any complex algebra 
A with a modulus e is the sum of its maximal nilpotent 
invariant sub-algebra N and a semi-simple algebra S 
which is a direct sum of t simple matric algebras Si. 
Then Si has the basal units e\& (a, /3 == 1, . . . . , pi), 
with 
(19) e U e hy =e U> <f}ei 5 =o{p9*y), e^e f yS =o(i^j), 
If v is an element of N such that 
then 
/ e i aa n ==n ì e k yy n—o (unless k = i, y= a), 
\ nefa = n, ne k yy =o (unless k =j, y = /3), 
and n is said to have the character 
(22) 
Let i and j be fixed integers such that e i 11 vef is not 
zero for every v in N and let v 1} v 2 , .... be elements of 
N such that