We consider generalizations of a well-known class of spaces, called $N\cup R$ by S. Mr\'{o}wka, where $R$ denotes an infinite maximal almost disjoint family (MADF) of countable subsets of the natural numbers $N$. We consider any uncountable cardinal $k$ and any MADF $A$ of countably infinite subsets of $k$. Let $\psi=\psi(k,A)$ denote the space with underlying set $\k\cup A$ and with the topology having as a base all singletons of $k$ and all sets of the form $\{A\}\cup (A- F)$ where $F$ is finite.

Mr\'{o}wka proved that there exists a MADF on $N$ such that the resulting $\psi$-space on $N$ with $R$ has a unique free $z$-ultrafilter $p_0$ (equivalently, the one-point compactification of $\psi$ and the Stone-Cech compactification of $\psi$ are the same). We extend this result of Mr\'{o}wka to uncountable cardinals. For cardinals at most the size of the cardinality of the continuum, we get an analogue of Mr\'{o}wka's theorem, but for cardinal greater than the cardinality of the continuum, it is always the case that there are many free $z$-ultrafilter on $\psi$. Yet there exists a special free $z$-ultrafilter $p$ on $\psi$ such that $p$ retains some of the properties that $p_0$ has in the countable case. In particular both $p$ and $p_0$ have a clopen local base in the Stone-Cech compactification of their respective $\psi$-spaces (although $\beta\psi$ need not be zero dimensional). A result for $p$ where $k$ is greater than the continuum, that $p_0$ does not share is that for certain $k$, $p$ will be a P-point in $\beta\psi$, but $p_0$ is not a P-point in its $\beta\psi$.