Commutative ring theory: Regular rings

Commutative rings I

Commutative Algebra, Vol. I. By O. Zariski and P. Samuel. Pp. xi 329. 52s. 6d. 1958. (D. van Nostrand, Princeton)

The divisor class group of a krull domain

Semistar Dedekind domains

The purpose of this article is to deepen the study of the relation between semistar operations on an integral domain D and the (semi) star operations (that is, the semistar operations, that restricted to the set of the fractional ideals, are star operations (on the overrings of D . First, we define the composition of two semistar operations and study when this composition is a semistar operation. Then we show that there is a bijection between the set of all semistar operations on a domain D and the set of all (semi) star operations on the overrings of D . To do this, we prove that semistar operations on D have a canonical decomposition as the composition of a semistar operation given by the extension to an overring and a (semi) star operation on this overring. Moreover, we study which properties of semistar operations are preserved by this bijection. Finally, we give some applications to the study of semistar operations on valuation and Prufer domains and we give, by using the techniques introduc...

/pdf/star-operations-on-overrings-and-semistar-operations-561nauhv9k.pdf

Star Operations on Overrings and Semistar Operations

A natural development of the recent work about semistar operations leads to investigate the concept of “semistar invertibility” Here, we show the existence of a “theoretical obstruction” for extending many results, proved for star-invertibility, to the semistar case For this reason, we introduce two distinct notions of invertibility in the semistar setting (called ⋆-invertibility and quasi-⋆-invertibility), we discuss the motivations of these “two levels” of invertibility and we extend, accordingly, many classical results proved for the d-, v-, t- and w-invertibility

Semistar invertibility on integral domains

After the introduction in 1994, by Okabe and Matsuda, of the notion of semistar operation, many authors have investigated different aspects of this general and powerful concept. A natural development of the recent work in this area leads to investigate the concept of invertibility in the semistar setting. In this paper, we will show the existence of a ``theoretical obstruction'' for extending many results, proved for star-invertibility, to the semistar case. For this reason, we will introduce two distinct notions of invertibility in the semistar setting (called $\star$--invertibility and quasi--$\star$--invertibility), we will discuss the motivations of these ``two levels'' of invertibility and we will extend, accordingly, many classical results proved for the $d$--, $v$--, $t$-- and $w$-- invertibility.

We study semistar Noetherian domains, that is, domains having the ascending chain condition on "quasi semistar ideals''.We generalize several of the classical results that hold in Noetherian domains to the case of semistar operations stable and of finite type, for instance, Cohen Theorem, primary decomposition, principal ideal Theorem, Krull intersection Theorem, etc. We do this mainly by using a method that allows one to transfer properties already proved for star operations to the context of semistar operations. Furthermore, an analogue of the Hilbert basis Theorem for semistar Noetherian domain (with respect to stable semistar operations) is proved.

/pdf/a-note-on-semistar-noetherian-domains-1dhaeth5yx.pdf

Giampaolo Picozza

Papers

Semistar Dedekind domains

Star Operations on Overrings and Semistar Operations

Semistar invertibility on integral domains

Semistar invertibility on integral domains

A note on semistar Noetherian domains