An Addendum to the Paper “Arithmetic Functions Over Rings with Zero Divisors”, Bull. Malaysian Math. Sc. Soc. 24 (2001), 81-91
Narkiewicz, Wladyslaw, and Ruengsinsub, Pattira, and Laohakosol, Vichian, (2004) An Addendum to the Paper “Arithmetic Functions Over Rings with Zero Divisors”, Bull. Malaysian Math. Sc. Soc. 24 (2001), 81-91. Bulletin of the Malaysian Mathematical Sciences Society, 27 (1). pp. 87-90. ISSN 0126-6705 Full text not available from this repository. Official URL: http://math.usm.my/bulletin/pdf/v27n1/v27n1p11.pdf AffiliationsWroclaw University, Institute of Mathematics
Kasetsart University, Dept. of Mathematics AbstractA commutative ring R, with unity and zero divisors, is a unique factorization ring, UFR for short, if for every non-zero non-unit r ∈ R , there exist irreducible elements r1,, rn such that r = r1 rn and whenever r = r1 rn = s1 sm (where r1,, rn , s1,, sm are irreducibles), then n = m and the s j can be renumbered so that ri is associated to si ( i = 1,, n) . Denote the set of arithmetic functions
over a ring R by AR , i.e. AR =  , and let Rω = R[[ x1, x2 ,]] and Rm = R[[x1,, xm ]] be the rings of formal power series in N0 , respectively, m indeterminates. It is well-known that AR and Rω are isomorphic. All UFR’s considered here throughout are assumed to have nonempty sets of zero divisors. In the cited paper, several results have been proved based on the hypothesis that “Rj is a UFR”. As observed by the first author, this hypothesis is void, as we now prove. Repository Staff Only: item control page
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