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The endomorphism algebras of modules of large cardinalities have been extensively studied in recent years using the combinatorial set-theoretic techniques of Shelah-the so-called black-box methods (see, e.g., [4, 5, 151). Despite the spectacular success of these methods, they are not suitable for realization theorems at small carinalities. Of course at the level of countability (or rather more generally for cardinals ~2’~) there are in some cases the original dramatic results of A. L. S. Corner [ 1, 2, 31 and the more recent generalizations of Gobel and May [ 111. Very recently the study of realization problems at cardinalities <2@’ has been relooked at in [S, 131 in relation to separable torsion-free abelian groups (and some generalizations to modules). In the latter paper a new type of support argument (which has roots in a much earlier work ) was introduced in an effort to circumvent the lack of a “black box.” It is this technique, which we exploit in the present paper to derive a basic realization result (Proposition 4), which can be readily adopted to, e.g., separable torsion-free abelian groups or modules over a complete discrete valuation ring. Among others we derive the following simple result, which appears to be new.
Gobel, R. & Goldsmith, B. (1991). On Separable Torsion-Free Modules of Countable Density Character. Journal of Algebra, vol. 144, no. 1, pg. 79-87. doi:10.1016/0021-8693(91)90128-U