  We study generalized comatrix coalgebras and upper triangular comatrix
coalgebras, which are not only a dualization but also an extension of classical
generalized matrix algebras. We use these to answer several questions on
Noetherian and Artinian type notions in the theory of coalgebras, and to give
complete connections between these. We also solve completely the so called
finite splitting problem for coalgebras: we show that a coalgebra $C$ has the
property that the rational part of every finitely generated left $C^*$-module
splits off if and only if $C$ has the form $C=\left(\begin{array}{cc} D & M \\
0 & E \end{array}\right)$, an upper triangular matrix coalgebra, for a serial
coalgebra $D$ whose Ext-quiver is a finite union of cycles, a finite
dimensional coalgebra $E$ and a finite dimensional $D$-$E$-bicomodule $M$.
