= Solution
The coefficient condition says precisely that every nonconstant monomial has positive powers of both variables. Thus
$$
R=K+XYK[X,Y].
$$
If $c+XYf$ and $d+XYg$ lie in $R$, so do their sum and their product
$$
cd+XY(cg+df+XYfg),
$$
so $R$ is a subring of $K[X,Y]$.
For $n\geq1$, let
$$
I_n=(XY,XY^2,\ldots,XY^n)_R.
$$
Then $I_n\subseteq I_{n+1}$. The containment is strict because $XY^{n+1}\notin I_n$: multiplying a generator $XY^j$ by an element of $R$ produces either a <scalar> multiple of $XY^j$, or terms divisible by $X^2$. It cannot produce the monomial $XY^{n+1}$ when $j\leq n$. Hence
$$
I_1\subsetneq I_2\subsetneq I_3\subsetneq\cdots
$$
is a strictly increasing <ideal> chain. The <constant-plus-ideal non-Noetherian subring> therefore satisfies
$$
\boxed{R\text{ is not Noetherian}}.
$$
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