On some classes of integral domains defined by Krull's $\boldsymbol{a.b.}$ operations

Details On some classes of integral domains defined by Krull's $\boldsymbol{a.b.}$ operations On some classes of integral domains defined by Krull's $\boldsymbol{a.b.}$ operations

2011-05-17

arxiv.org/abs/1105.3319v1

Mathematics

Commutative Algebra

Scientific paper

Let $D$ be an integral domain with quotient field $K$. The $b$-operation that associates to each nonzero $D$-submodule $E$ of $K$, $E^b := \bigcap\{EV \mid V valuation overring of D\}$, is a semistar operation that plays an important role in many questions of ring theory (e.g., if $I$ is a nonzero ideal in $D$, $I^b$ coincides with its integral closure). In a first part of the paper, we study the integral domains that are $b$-Noetherian (i.e., such that, for each nonzero ideal $I$ of $D$, $I^b = J^b$ for some a finitely generated ideal $J$ of $D$). For instance, we prove that a $b$-Noetherian domain has Noetherian spectrum and, if it is integrally closed, is a Mori domain, but integrally closed Mori domains with Noetherian spectra are not necessarily $b$-Noetherian. We also characterize several distinguished classes of $b$-Noetherian domains. In a second part of the paper, we study more generally the e.a.b. semistar operation of finite type $\star_a$ canonically associated to a given semistar operation $\star$ (for instance, the $b$-operation is the e.a.b. semistar operation of finite type canonically associated to the identity operation). These operations, introduced and studied by Krull, Jaffard, Gilmer and Halter-Koch, play a very important role in the recent generalizations of the Kronecker function ring. In particular, in the present paper, we classify several classes of integral domains having some of the fundamental operations $d$, $t$, $w$ and $v$ equal to some of the canonically associated e.a.b. operations $b$, $t_a$, $w_a$ and $v_a$.

Affiliated with

Also associated with

No associations

Rating

On some classes of integral domains defined by Krull's $\boldsymbol{a.b.}$ operations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On some classes of integral domains defined by Krull's $\boldsymbol{a.b.}$ operations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On some classes of integral domains defined by Krull's $\boldsymbol{a.b.}$ operations will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-727719