An Introduction to Nonassociative Algebras by Richard D. Schafer

Author:Richard D. Schafer [Schafer, Richard]
Language: eng
Format: epub, pdf
Publisher: Dover Publications

If is a proper ideal of , then .

Proof. Since is an ideal of , we have either or by (ii). In the first case . But then by (iii), implying , a contradiction. Hence . Let be the image of under the projection of onto relative to the direct sum decomposition , that is,

Then is a right ideal of . For s in in , imply there exists t in such that s + t in , so contains (s + t)s′ = ss′ + ts′ where ss′ is in , ts′ in by (iii); hence ss′ is in as desired. Similarly is a left ideal of . Hence or . The first case leads to a contradiction. For implies that, for every s in , there exists t in such that s + t is in . Let s1,..., sn be a basis for over F. Then there exist t1,..., tn in , satisfying . Hence there is a linear mapping U of into (extending siU = tt) which satisfies

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