It is an important problem in group theory to determine whether or not direct summands of groups in a class belong again to the same class. The problem is particularly challenging for the class of cartesian powers $\doubz\sp\kappa$ of the group of integers $\doubz$ ($\kappa$ denotes a cardinal). In the absence of $\omega$-measurable cardinals the problem is answered in the affirmative. But there are few results to date on direct sum decompositions of $\doubz\sp\kappa$ where $\kappa$ denotes an $\omega$-measurable cardinal In this dissertation we construct examples of direct sum decompositions of cartesian powers $\doubz\sp\kappa$ = $\prod\sb{i\in\kappa}\langle e\sb{i}\rangle$ of the group of integers $\doubz$ where $\kappa$ denotes an $\omega$-measurable cardinal, and $\langle e\sb{i}\rangle\cong\doubz$ for each $i\in\kappa$. The decompositions have the following form:$$\doubz\sp\kappa=A\oplus B,\ {\rm such\ that}\ e\sb{i}\in A\ {\rm for\ all}\ i\in\kappa$$ We introduce the notion of completely independent ultrafilters, and use that concept to derive sufficient conditions for a summand of $\doubz\sp\kappa$ to be isomorphic to $\doubz\sp\lambda$ for some cardinal $\lambda.$ Furthermore we show that summands of $R\sp\mu$, where $R<\doubq$ is a rational group and $\mu$ denotes the first measurable cardinal, are necessarily again powers of the rational group R