Abstract:
Let R be a commutative ring with I. R is called a P.P. ring if every principal ideal of R is projective as an R - module and R is called semi-hereditary if every finitely generated ideal of R is projective as an R - module. [4]. Di. A ring R is called a valuation ring if for all a, b ε R, a b or b a. Endo in [4] showed that R is a P.P. ring iff the total qoutient ring Rs of R is regular (in Von Neumann sence and Rmis an integral domain for each maximal ideal M or R. He also proved in [5] that I is semi-hereditary iff the total quotient ring of R is regular and Rmis a valuation domain. In $ 1 of this paper we give a new proof of this theorem based on the first theorem. Moreover we give other characterizations of semi-hereditary rings in case R is a P.P. ring (see Theorem 1.8). In $ 2 we give another characterization based on the concept of p -projective ideals. We say an ideal I is p -projective if there exists a positive integar n such that 1n is projective as an R-module. We say that R is p -semi-hereditary if every finitely generated ideal of R is p -projective.
Page(s):
119-124
DOI:
DOI not available
Published:
Journal: Islamabad Journal of Science, Volume: 7, Issue: 1--2, Year: 1980