We consider a total matric algebra M over a field F, whose general element is u = Yj a a e ih (h j = 1> * ' * > n )> where e^^ = ea if j = 1, and e^en = 0 for j^L THEOREM 1. A necessary and sufficient condition that u = ^La^e^ be idempotent in M is This is seen immediately on writing U ~2!_jO£psOtsqepq P,Q,s and comparing with U / jOLygepa» P,Q THEOREM 2. A necessary and sufficient condition f or an idempotent element u to be primitive in M is (2) a pi a jq = a pg (Xji, (p, q, i, j = 1, • •

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... . For, let u=^2a r te r t r,t be a primitive indempotent element of M. Let a^^O. Then the element ue^u/aji is idempotent in uMu, since ( ue%jU\ u' C{jUCij' iA> UOLj%eîjU ue%ju otji ) OL}I a]i an Hencef we have ueiju/ctji = u, and ue^u -a^u. Equating coef-* Presented to the Society, June 15, 1927. t Dickson, Algebras and their Arithmetics, p. 55.