... C, we can then write A = B ⊕ C, the direct sum of B and C as algebras.As in the case of vector spaces, the notion of direct sum extends to an arbitrary (indexed) set of summands. In these notes ... 2n+1elements of B, no matter how associated,there is a final multiplication which is performed. At least one of the two factors is a product of at least 2nelements of B, while the otherfactor bis ... S1· · · St of t elements in B∗is 0, any product of 2telements of B, no matter how associated, is 0. That is, B is nilpotent. The Project Gutenberg EBook of An Introduction to Nonassociative...