×

zbMATH — the first resource for mathematics

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
PDF BibTeX Cite
Full Text: DOI
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.