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Conjugacy classes of left ideals of a finite dimensional algebra

Arkadiusz Mȩcel and Jan Okniński

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Let $A$ be a finite dimensional unital algebra over a field $K$ and let $C(A)$ denote the set of conjugacy classes of left ideals in $A$. It is shown that $C(A)$ is finite if and only if the number of conjugacy classes of nilpotent left ideals in $A$ is finite. The set~$C(A)$ can be considered as a semigroup under the natural operation induced from the multiplication in $A$. If $K$ is algebraically closed, the square of the radical of~$A$ is zero and $C(A)$ is finite, then for every $K$-algebra $B$ such that $C(B)\cong C(A)$ it is shown that $B\cong A$.

Article information

Publ. Mat., Volume 57, Number 2 (2013), 477-496.

First available in Project Euclid: 12 December 2013

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Zentralblatt MATH identifier

Primary: 16P10: Finite rings and finite-dimensional algebras {For semisimple, see 16K20; for commutative, see 11Txx, 13Mxx} 16D99: None of the above, but in this section 20M99: None of the above, but in this section

Finite dimensional algebra left ideal semigroup conjugacy class


Mȩcel, Arkadiusz; Okniński, Jan. Conjugacy classes of left ideals of a finite dimensional algebra. Publ. Mat. 57 (2013), no. 2, 477--496.

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