  In this paper we explore the almost Cohen-Macaulayness of the associated
graded ring of stretched $\mathfrak{m}$-primary ideals with small first Hilbert
coefficient in a Cohen-Macaulay local ring $(A,\mathfrak{m})$. In particular,
we explore the structure of stretched $\mathfrak{m}$-primary ideals satisfying
the equality $\mathrm{e}_1(I)=\mathrm{e}_0(I)-\ell_A(A/I)+4$ where
$\mathrm{e}_0(I)$ and $\mathrm{e}_1(I)$ denote the multiplicity and the first
Hilbert coefficient respectively.

Commutative Algebra