Abstract
Let
({\mathrm{\Gamma}},\le )
be a strictly ordered monoid, and let
{{\mathrm{\Gamma}}}^{\ast }\left={\mathrm{\Gamma}}\backslash \{0\}
. Let
D\subseteq E
be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D. Set
\begin{array}{l}D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {E}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(0)\in D\right\}\hspace{.5em}\text{and}\\ \hspace{0.2em}D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {D}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(\alpha )\in I,\hspace{.5em}\text{for}\hspace{.25em}\text{all}\hspace{.5em}\alpha \in {{\mathrm{\Gamma}}}^{\ast }\right\}.\end{array}
In this paper, we give necessary conditions for the rings
D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt]
to be Noetherian when
({\mathrm{\Gamma}},\le )
is positively ordered, and sufficient conditions for the rings
D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt]
to be Noetherian when
({\mathrm{\Gamma}},\le )
is positively totally ordered. Moreover, we give a necessary and sufficient condition for the ring
D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt]
to be Noetherian when
({\mathrm{\Gamma}},\le )
is positively totally ordered. As corollaries, we give equivalent conditions for the rings
D+({X}_{1},\ldots ,{X}_{n})E{[}{X}_{1},\ldots ,{X}_{n}]
and
D+({X}_{1},\ldots ,{X}_{n})I{[}{X}_{1},\ldots ,{X}_{n}]
to be Noetherian.
Pontryagin’s Maximum Principle for Optimal Control of Stochastic SEIR Models
