Abstract. Let $R$ be a left slender ring and let $\kappa $ and $ \alpha $ be infinite cardinal numbers. Suppose $\phi\colon R^{\kappa }\rightarrow R^{< \alpha }$ is a left $R$-module homomorphism, where $R^{< \alpha }$ consists of all elements $x$ in $R^{\alpha }$ such that $\left |\mathop{\rm supp} (x)\right | < \alpha $. Using new set-theoretic techniques we will show that, if $\left | R\right | < \mu $, the least measurable cardinal number, then the image of $\phi $ is contained in a copy of $R^{\gamma }$, where $\gamma < \alpha $ if $\alpha\leq \mu $ as well as when $\kappa < \mu $.

AMS Subject Classification (1991): 16D80, 20K25; 13C13, 20K30

Received April 2, 2001, and in revised form July 5, 2002. (Registered under 2881/2009.)