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CHAPTER VIII 
THE PRINCIPAL THEOREM ON ALGEBRAS 
74. Introduction. We shall prove that any associ 
ative algebra over a non-modular field F is either semi- 
simple or the sum of its maximal nilpotent invariant 
sub-algebra and a semi-simple algebra, each over F. 
For the special case in which F is the field of all complex 
numbers, a more elementary proof is given in § 79. 
We shall need to employ extensions of the given field 
F. In this connection, note that the theorem of § 66 
implies the 
Corollary. Let A he an algebra over a non-modular 
field F. Let F x denote any field containing F as a sub 
field. Denote by A i the algebra over F z which has the same 
basal units* {and hence the same constants of multiplica 
tion) as algebra A over F. Then A l is semi-simple if and 
only if A is semi-simple. But if A has a maximal nil- 
potent invariant sub-algebra N, that of A x is the algebra over 
F x which has the same basal units as N. 
75. Direct product of simple matric algebras. Let A 
be a simple matric algebra over F with the m 2 basal 
units a { j such that (§51) 
(1) aija rs — o (j7 1 — r)j aijaj s ai s (f, j, r t s 1, . ... , m). 
Let B be a simple matric algebra over F with n 3 basal 
units b rs {r,s=i, . . . . , n), satisfying relations of 
* They may be assumed to be linearly independent with respect to 
Fi by § 13.