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Embedding nil algebras in train algebras. (English) Zbl 0802.17026
Let $$N$$ be a right nil algebra of index $$n\geq 2$$ over a field $$F$$ and $$r\in F$$. We construct a right train algebra $$(A,\omega)$$ with a nonzero idempotent $$e\in A$$, such that $$\text{Ker } \omega=N$$ and $$ea=ae= ra$$, for all $$a\in N$$. This algebra has rank $$n+1$$ if $$2r\neq 1$$ or $$n$$ if characteristic of $$F$$ is not 2 and $$r= 1/2$$. This is a generalization of the classical example, due to V. M. Abraham [Proc. Edinb. Math. Soc., II. Ser. 20, 53-58 (1976; Zbl 0361.17007)], of a train algebra that is not special train, to not necessarily commutative or associative right nil algebras of index $$n$$.

##### MSC:
 17D92 Genetic algebras
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##### References:
 [1] Wörz, Algebras in Genetics 36 (1980) [2] Suttles, Notices Amer. Math. Soc. A19 pp 566– (1972) [3] Abraham, Proc. Edinburgh Math. Soc. 20 pp 53– (1976) [4] Etherington, Proc. Roy. Soc. Edinburgh 59 pp 242– (1939) · Zbl 0027.29402 [5] DOI: 10.1016/0024-3795(93)90434-P · Zbl 0811.17034 [6] DOI: 10.1007/BF01194568 · Zbl 0616.17011
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