Associative Algebras with a Distributive Lattice of Subalgebras
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Algebra and Logic, Vol. 59, No. 5, November, 2020 (Russian Original Vol. 59, No. 5, September-October, 2020)
ASSOCIATIVE ALGEBRAS WITH A DISTRIBUTIVE LATTICE OF SUBALGEBRAS A. G. Gein∗
UDC 512:522.3
Keywords: lattice of subalgebras, distributive lattice, lattice of subextensions of field. We give a full description of associative algebras over an arbitrary field, whose subalgebra lattice is distributive. All such algebras are commutative, their nil-radical is at most two-dimensional, and the factor algebra with respect to the nil-radical is an algebraic extension of the base field. The paper of Ore [1] containing a description of groups with a distributive lattice of subgroups stimulated similar investigations in semigroup theory [2], associative [3] and Lie [4] rings, etc. However, this issue remained open for associative algebras over fields. The lattice of subalgebras of an associative algebra is “poorer” than the lattice of subrings of an associative ring; therefore, the results for algebras cannot be obtained by reducing the corresponding results for rings. Throughout what follows, A is an associative algebra over a field F with a distributive lattice of subalgebras. Clearly, the subalgebra lattice of any subalgebra of the algebra A and any of its factor algebras is also distributive. We use X to denote a subalgebra generated by a set X; if X = {a}, we write a. Let B ∨C be a subalgebra generated by subalgebras B and C of A. The direct sum of subalgebras B and C as linear spaces is denoted by B ⊕ C; the direct sum of two-sided ideals B and C is denoted by B × C. In what follows, two-sided ideals are called ideals for simplicity. An element a of an associative algebra is said to be algebraic if a is a finite-dimensional subalgebra. This condition is equivalent to the existence of a nonzero polynomial f (λ) without a free term with coefficients in F for which f (a) = 0. Recall that an element a is nilpotent if there ∗
Supported through the Competitiveness Project (Agreement No. 02.A03.21.0006 of 27.08.2013 between the Ministry of Education and Science of the Russian Federation and the Ural Federal University).
El’tsin Ural Federal University, Yekaterinburg, Russia; [email protected]. Translated from Algebra i Logika, Vol. 59, No. 5, pp. 517-528, September-October, 2020. Russian DOI: 10.33048/alglog.2020.59.501. Original article submitted January 18, 2020; accepted November 27, 2020. c 2020 Springer Science+Business Media, LLC 0002-5232/20/5905-0349
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exists a natural number n for which an = 0. The least such number is called the nilpotency index of an element a. In what follows, nilpotent elements are called nil-elements for brevity. Recall also that an element e for which e2 = e is called an idempotent. It is well known that each finite-dimensional associative algebra contains either a nonzero nil-element or a nonzero idempotent (possibly, both). A zero subalgebra is the smallest element of the lattice of subalgebras of an arbitrary associative algebra. An element that covers the smallest element of the
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