§ 76] DIVISION ALGEBRA AS DIRECT SUM 
121 
If each Di is of order i, our theorem holds for F 1 =F'. 
In the contrary case, we employ an extension F" of 
F' such that the algebra over F", having the same 
ni{fii> i) basal units as D i} is not a division algebra. 
To it we apply the argument just made for D'. 
Since the division algebras introduced at any stage 
are all of orders less than those of the preceding stage, 
the process terminates, so that we reach a final stage in 
which the division algebras are all of order i. Each 
division algebra of the prior stage is therefore a direct 
sum of simple matric algebras. Our theorem now follows 
from that in § 75. 
77. Theorem.* If A is an algebra having a single 
idem potent element e over a non-modular field F, then A can 
he expressed in the form A = B-\-N, where B is a division 
algebra and N is zero or the maximal nilpotent invariant 
sub-algebra of A. 
The theorem is obvious when A is of order 1, since 
then A =A +0 and A is a division algebra. 
To prove the theorem by induction, assume it for all 
algebras of type A which are of orders less than the order 
of A, 
We first show that we may take N 2 = o. Let N 2 ^o 
and write 
(4) A=B'+N, B'y\N = o, iV=iVx+iV 2 , N1 y\N 2= o. 
Since AN 2 =AN • N^N • N and N 2 A SN 2 , N 2 is an in 
variant sub-algebra of A. 
The classes f (x) of A modulo N 2 are the elements of 
A—N 2 . In particular, the classes {nf), each uniquely 
* In § 79 there is a far simpler proof for the case of algebras A over 
the field of all complex numbers. 
t The notation (x) marks the distinction from classes [x] modulo N.