176 ARITHMETIC OF AN ALGEBRA- [chap, x 
Evidently, 
A=B Z +B 2 + .... +B a , B P B q SA p A q . 
Now A p+q is B a if ; but, for p+q<a, 
A p + q =B p+q +A p + q +' = B p+q +B P+q+1 +A p + q + 2 
= . , . . =-£>0-(- 9 + .... +-S a . 
We now assume that T is nilpotent and of index a, 
so that B a = o. Let n z , .... , nb 1 be a basis of B z , 
i.e., linearly independent elements of B z such that every 
element of B z is a linear combination of them with 
coefficients in the field F over which A is defined. Let 
Ubj.+x, . , nf >i +b 2 be a basis of B 2 , etc. 
First, let pSq and p+q<a. Then in the bases of 
B p and B q , each n has a subscript ^b z -\- ... . -\-b q . 
The latter sum is less than the minimum subscript 
&!+.... A-b P + q - 1 +1 of an n in B p + q . Hence by 
(14), every product fiifij is a linear combination with coeffi 
cients in F of those n’s whose subscripts exceed both i and j. 
The same result holds also if w* is in B P and nj is in 
B q , where now p+q^a, since B P B q =o by (15), so that 
niUj = 0. 
A set of basal units n z , n 2 , .... of a nilpotent 
algebra is called a normalized set if it has the property 
expressed in italics. 
101. The two categories of complex algebras. By 
§ 79, every complex algebra A with a modulus e is the 
sum of its maximal nilpotent invariant sub-algebra N 
and a semi-simple sub-algebra S, while S is a direct sum 
of simple matric algebras Si. Here N must be replaced 
by o if A itself is semi-simple. According as the orders