Formal power series
E.D. Cashwell, Cath Everett · Pacific Journal of Mathematics · 1963
Introduction* It has been shown [2] that the set Ω of all arithmetic functions a on N = {1,2, 3, •} to the complex field C is a unique factorization domain under ordinary addition and the "arithmetic" product:The proof was based on the obvious isomorphism between Ω and the domain C[[x lf x 2 , •]] of formal power series over C, in countably many variables, induced by the mapping a and the fact that the domain C [[x 19 , x n ]] of such series in any finite number of variables is factorial (i.e., a unique factorization domain).Recently D. Buchsbaum [1] and P. Samuel [8] have shown that the latter domains are factorial whenever C is a regular factorial domain, in particular, a principal ideal ring.It therefore seems appropriate to generalize our previous result in the following way.We replace the integers n, which in standard form are uniquely defined by their sequences a u a 2 , of exponents, by vectors α, finitely nonzero on an arbitrary set I of indices i, and consider the corresponding ring Ω of functions defined on the set of all such vectors to an arbitrary domain of integrity C.The analogue of the above mapping establishes an isomorphism between Ω and the ring P of formal power series over C in the variables x if iel.We prove that these rings, Ω and P, are factorial if and only if all domains C[[x 19 ••-,#"]] are factorial.It follows from the theorem of Buchsbaum and Samuel that P is factorial whenever the coefficient domain is a regular factorial domain.Specializing to the case of a countable set of indices, we see that the ring of integer-valued arithmetic functions is factorial (cf.[9]; p. 36).Indeed this is true if the complex field C is narrowed to the integral domain R[Θ] of any algebraic number field provided R[Θ] is itself factorial [7; p. 99].These results appear as corollaries of a general theorem on Gaussian semi-groups to which we devote the second part of the paper.1.The ring Ω of functions* Let I be an arbitrary infinite set of