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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 \(n\).

17D92 Genetic algebras
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[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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