  Let $R:= K[x_1,\ldots,x_{n}]$ be a polynomial ring over an infinite field
$K$, and let $I \subset R$ be a homogeneous ideal with respect to a weight
vector $\omega = (\omega_1,\ldots,\omega_n) \in (\mathbb{Z}^+)^n$ such that
$\dim(R/I) = d$. In this paper we study the minimal graded free resolution of
$R/I$ as $A$-module, that we call the Noether resolution of $R/I$, whenever $A
:=K[x_{n-d+1},\ldots,x_n]$ is a Noether normalization of $R/I$. When $d=2$ and
$I$ is saturated, we give an algorithm for obtaining this resolution that
involves the computation of a minimal Gr\"obner basis of $I$ with respect to
the weighted degree reverse lexicographic order. In the particular case when
$R/I$ is a $2$-dimensional semigroup ring, we also describe the multigraded
version of this resolution in terms of the underlying semigroup. Whenever we
have the Noether resolution of $R/I$ or its multigraded version, we obtain
formulas for the corresponding Hilbert series of $R/I$, and when $I$ is
homogeneous, we obtain a formula for the Castelnuovo-Mumford regularity of
$R/I$. Moreover, in the more general setting that $R/I$ is a simplicial
semigroup ring of any dimension, we provide its Macaulayfication.
  As an application of the results for $2$-dimensional semigroup rings, we
provide a new upper bound for the Castelnuovo-Mumford regularity of the
coordinate ring of a projective monomial curve. Finally, we describe the
multigraded Noether resolution and the Macaulayfication of either the
coordinate ring of a projective monomial curve $\mathcal{C} \subseteq
\mathbb{P}_K^{n}$ associated to an arithmetic sequence or the coordinate ring
of any canonical projection $\pi_{r}(\mathcal{C})$ of $\mathcal{C}$ to
$\mathbb{P}_K^{n-1}$.
