On semi-hereditary rings

Shizuo ENDO
1961 Journal of the Mathematical Society of Japan  
A ring R with unit element is called " left (right) semi-hereditary " according to [21 if any finitely generated left (right) ideal of R is projective. The purpose of this paper is to determine completely the structure of commutative semi-hereditary rings. A. Hattori has recently given in [6] a homological characterization of Prefer rings, i. e., semi-hereditary integral domains. This was generalized by M. Harada [51 to commutative rings whose total quotient rings are regular. The results of
more » ... s paper will include those results of [5] and [61. In § 3 we shall give a necessary and sufficient condition for a ring to be regular by using the quotient rings. Also we shall introduce a notion of quasiregular rings and show some properties of them. In § 4 we shall characterize semi-hereditary rings by using the quotient rings as follows : A ring R is semi-hereditary if and only if the total quotient ring K of R is regular and the quotient ring Rm of R with respect to any maximal ideal m of R is a valuation ring. Furthermore we shall introduce a notion of algebraic extensions of regular rings and show that the integral closure R' of a semi-hereditary ring R in any algebraic extension K' of the total quotient ring K of R is also semi-hereditary. In § 5, we shall first prove that a local ring R is a valuation ring if and only if w. gl. dim R <_ 1. Secondly we shall show, as a generalization of [6], Theorem 2, that a ring R with the total quotient ring K is semi-hereditary if and only if w. gl. dim R <_ 1 and w. gl. dim K= 0, or if and only if any torsionfree R-module is flat. § 2. Notations and terminologies. Throughout this paper a ring will mean a commutative ring with unit element 1. Our notations and terminologies are, in general, the same as in [2] but we shall make the following modifications. A local ring will mean a (not always Noetherian) ring with only one
doi:10.2969/jmsj/01320109 fatcat:yp2yo2b2ubhkhkb3un2qsl3fqu