On homogeneous mappings of mixed modules

D. S. Chistyakov

Čebyševskij sbornik, Tome 18 (2017) no. 2, pp. 256-266

Résumé

In this paper we study mixed modules, with the following property: every homogeneous function of several variables of a module is additive. By a homogeneous function we mean any mapping of the direct sum of a finite number of copies of a module into the module itself that commutes with the endomorphisms of the given module. In the universal algebra, the algebraic structure is said to be endoprimal if all its term-functions commute with endomorphisms. It is well-known that each endodualizable finite algebra is endoprimal. Some authors have studied endoprimal algebras in varieties of vector spaces, semilattices, Boolean algebras, Stone algebras, Heyting algebras, and Abelian groups. In this article, the links between endoprimality and the properties of the multiplicative semigroup of the endomorphism ring of a module, which the author started earlier. Classes of mixed non-reduced splitting modules and reduced modules over commutative Dedekind ring have been investigated. Links between this problem and the property of unique additivity has been shown. Bibliography: 26 titles.

Détail

Keywords: Dedekind ring, divisible module, reduced module, mixed module, homogeneous map, term-function, endofunction.