§ 77] ALGEBRA WITH SINGLE IDEMPOTENT 
123 
It remains to prove the theorem when N 2 — o, a 
property utilized only at the end of the proof. 
By § 38, D=A — N is semi-simple and has a modulus. 
It has no other idempotent element since A has a single 
one. Hence by Corollary 1 of § 43, D is a division 
algebra. 
By § 76, we may extend the initial field to a field F x 
such that the algebra D z over F z , which has the same 
basal units as D, is a direct sum of simple matric algebras. 
Denote by A z and N x the algebras over F x which have 
the same basal units as A and N, respectively. By 
§ 74, N x is the maximal nilpotent invariant sub-algebra 
of Aj. Hence A Z — N 1 = D Z . 
By § 54; -d 1 contains a sub-algebra C equivalent to 
A x —N x , whence A 1 = C-\-N 1 , C^N z = o. Let e z , . , 
e c be a set of basal units of C. Since A—N is of order 
c, the basal units of N (or N x ) together with certain c 
elements a z , . . . . , a c of A form a set of basal units of 
A (or Aj). Hence we may write 
C 
(5) 
C) 
where the n{ are elements of N z , and the a# are numbers 
of F x whose determinant is not zero (otherwise, as in 
§ 5, a linear combination of e z , . . . . , e c would belong 
to N z , contrary to Cs\N 1 = o). Solving (5), we get 
(6) 
where the 18# are in F z and their determinant is not 
zero. Write