On f-Prüfer Rings and f-Bezout Rings

• Autores: David F. Anderson
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 30, Nº 2, 2004, págs. 331-343
• Idioma: inglés
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• Resumen

  • The purpose of this paper is to introduce two new classes of rings that are closely related to the classes of Prüfer domains and Bezout domains. Let R be a commutative ring with 1 such that Nil(R) (the ideal of nilpotent elements of R) is a divided prime ideal of R, T(R) be the total quotient ring of R, and Z(R) be the set of zerodivisors of R. Then the map phi from T(R) into R_Nil(R) defined by phi(a/b) = a/b for every a in R and b in R\Z(R) is a ring homomorphism from T(R) into R_Nil(R), and phi restricted to R is also a ring homomorphism from R into R_Nil(R) given by phi(x) = x/1 for every x \in R. A nonnil ideal I of R is said to be phi-invertible if phi(I) is an invertible ideal of phi(R). If every finitely generated nonnil ideal of R is phi-invertible, then we say that R is a phi-Prüfer ring. Also, we say that R is a phi-Bezout ring if phi(I) is a principal ideal of phi(R) for every finitely generated nonnil ideal I of R. We show that the theories of phi-Prüfer and phi-Bezout rings resemble that of Prüfer and Bezout domains.