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The first two Hilbert coefficients of a primary ideal play an important role in commutative algebra and in algebraic geometry. In this paper we give a complete algebraic structure of the Sally module of integrally closed ideals
$I$
in a Cohen–Macaulay local ring
$A$
satisfying the equality
$\text{e}_{1}(I)=\text{e}_{0}(I)-\ell _{A}(A/I)+\ell _{A}(I^{2}/QI)+1,$
where
$Q$
is a minimal reduction of
$I$
, and
$\text{e}_{0}(I)$
and
$\text{e}_{1}(I)$
denote the first two Hilbert coefficients of
$I,$
respectively, the multiplicity and the Chern number of
$I.$
This almost extremal value of
$\text{e}_{1}(I)$
with respect to classical inequalities holds a complete description of the homological and the numerical invariants of the associated graded ring. Examples are given.
Let (A,m) be a Noetherian local ring with d = dim A ≥ 2. Then, if A is a Buchsbaum ring, the first Hilbert coefficients 
of A for parameter ideals Q are constant and equal to 
where hi(A) denotes the length of the ith local cohomology module 
of A with respect to the maximal ideal m. This paper studies the question of whether the converse of the assertion holds true, and proves that A is a Buchsbaum ring if A is unmixed and the values are constant, which are independent of the choice of parameter ideals Q in A. Hence, a conjecture raised by [GhGHOPV] is settled affirmatively.
