  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 $v_a$.
