Koreshkov, N. A. Modules and ideals of algebras of associative type. (English. Russian original) Zbl 1241.17002 Russ. Math. 52, No. 8, 20-27 (2008); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2008, No. 8, 25-34 (2008). Summary: We study some properties of algebras of associative type introduced in previous papers of the author [Russ. Math. 50, No. 9, 32–39 (2006); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2006, No. 9, 34–42 (2006), Russ. Math. 51, No. 3, 33–41 (2007); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2008, No. 3, 38–46 (2007; Zbl 1211.17003)]. We show that a finite-dimensional algebra of associative type over a field of zero characteristic is homogeneously semisimple if and only if a certain form defined by the trace form is nonsingular. For a subclass of algebras of associative type, it is proved that any module over a semisimple algebra is completely reducible. We also prove that any left homogeneous ideal of a semisimple algebra of associative type is generated by a homogeneous idempotent. Cited in 1 ReviewCited in 2 Documents MSC: 17A30 Nonassociative algebras satisfying other identities Keywords:algebra of associative type; homogeneously semisimple algebra; module; ideal; homogeneous idempotent Citations:Zbl 1211.17003 PDFBibTeX XMLCite \textit{N. A. Koreshkov}, Russ. Math. 52, No. 8, 20--27 (2008; Zbl 1241.17002); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2008, No. 8, 25--34 (2008) Full Text: DOI References: [1] Y. Bahturin and M. Zaicev, ”Identities of Graded Algebras,” J. Algebra. 205(1), 1–12 (1998). · Zbl 0920.16011 [2] Y. Bahturin, M. Zaicev, and S. K. Sehgal, ”G-Identities of Nonassociative Algebras,” Matem. Sborn. 190(11), 3–14 (1999). [3] N. A. Koreshkov, ”On Nilpotency and Decomposition of Algebras of Associative Type,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 9, 34–42 (2006) [Russian Mathematics (Iz. VUZ) 50 (9), 32-39 (2006)]. [4] N.A. Koreshkov, ”A Class of Algebras of Associative Type,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 3, 38–46 (2007) [Russian Mathematics (Iz. VUZ) 51 (3), 33-41 (2007)]. · Zbl 1211.17003 [5] N. G. Chebotaryov, Introduction to the Theory of Algebras (OGIZ, Gostekhizdat, Moscow-Leningrad, 1949) [in Russian]. [6] I. Herstein, Noncommutative Rings (Wiley, NY, 1968; Mir, Moscow, 1972). · Zbl 0177.05801 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.