1964/06/01 by George M. Bergman · 37 citations
Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Combinatorics #Commutative property #Commutative ring #Discrete mathematics #Endomorphism #Geometry #Ideal (ethics) #Law #Left and right #Mathematical proof #Mathematics #Multiplication (music) #Prime (order theory) #Primitive ring #Primitive root modulo n #Principal ideal ring #Pure mathematics #Ring (chemistry) #Rings, Modules, and Algebras
paper · doi:10.1090/s0002-9939-1964-0167497-4
published in Proceedings of the American Mathematical Society 15(3), 473-475 (American Mathematical Society)
openalex publication_date 1964/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/03/11
Jacobson [1, p. 4] writes, is not known whether [right] primitivity implies left primitivity. It seems unlikely that it does, but no examples of [right] primitive rings which are not left primitive are known. Such an example is here constructed. Let D be a division ring, and a: D -D an endomorphism. Let A be the ring of formal polynomials Eizo diYi (dIED, nonzero for only finitely many i) with multiplication determined by the rule Yd =a(d) Y. Such rings are without zero divisors, and every left ideal of one is principal. The proofs are exactly as for the ordinary commutative rings of polynomials, cf. [2, p. 483]. Now let D = Q(X), where Q denotes the field of rationals, and let a be the map r(X)->r(X2). In this case, we have the following result.