Finitistic weak dimension of commutative arithmetical rings

A local ring O is called regular if every finitely generated ideal I ◃ O possesses finite projective dimension. In the article localizations O = Aq, q ∈ Spec A, of a finitely presented, flat algebra A over a Prüfer domain R are investigated with respect to regularity: this property of O is shown to be equivalent to the finiteness of the weak homological dimension wdim O. A formula to compute wdim O is provided. Furthermore regular sequences within the maximal ideal M ◃ O are studied: it is shown that regularity of O implies the existence of a maximal regular sequence of length wdim O. If q ∩ R has finite height, then this sequence can be chosen such that the radical of the ideal generated by its members equals M. As a consequence it is proved that if O is regular, then the factor ring O/(q ∩ R)O, which is noetherian, is Cohen–Macaulay. If in addition (q ∩ R)Rq ∩ R is not finitely generated, then O/(q ∩ R)O itself is regular.

Download Full-text

WEAK DIMENSION AND CHAIN-WEAK DIMENSION OF ORDERED SETS

Bulletin of the Korean Mathematical Society ◽

10.4134/bkms.2005.42.2.315 ◽

2005 ◽

Vol 42 (2) ◽

pp. 315-326

Author(s):

JONG-YOUL KIM ◽

JEH-GWON LEE

Keyword(s):

Ordered Sets ◽

Weak Dimension

Download Full-text

REGULARITY AND STRONG REGULARITY IN THE CONTEXT OF CERTAIN CLASSES OF RINGS

Journal of Algebra and Its Applications ◽

10.1142/s0219498812502052 ◽

2013 ◽

Vol 12 (05) ◽

pp. 1250205 ◽

Cited By ~ 1

Author(s):

MICHAŁ ZIEMBOWSKI

Keyword(s):

Positive Integer ◽

Sufficient Condition ◽

Strong Regularity ◽

Necessary And Sufficient Condition ◽

Weak Dimension ◽

Ring Of Polynomials ◽

Necessary And Sufficient ◽

Semihereditary Rings ◽

The Ideal

We consider the ring R[x]/(xn+1), where R is a ring, R[x] is the ring of polynomials in an indeterminant x, (xn+1) is the ideal of R[x] generated by xn+1 and n is a positive integer. The aim of this paper is to show that regularity or strong regularity of a ring R is necessary and sufficient condition under which the ring R[x]/(xn+1) is an example of a ring which belongs to some important classes of rings. In this context, we discuss distributive rings, Bézout rings, Gaussian rings, quasi-morphic rings, semihereditary rings, and rings which have weak dimension less than or equal to one.

Download Full-text