Unreduced generalized endoprimal abelian groups

The endofunction on abelian group $A$ is the function $f: A^n\to A$, such that $\varphi f(x_1,\ldots, $ $ x_n) = f(\varphi(x_1),\ldots, \varphi(x_n))$ for all endomorphisms $\varphi$ of group $A$ and all $n $ from $ \mathbb{N}$. If each endofunction has the form $f(x_1,\ldots, x_n) = \sum_{i = 1}^n \lambda_ix_i$ for some central endomorphisms $\lambda_1,\ldots, \lambda_n$ of a group $A$, then such a group is called generalized endoprimal ($GE$-group). In the paper, we find $GE$-groups in the class of nonreduced abelian groups. In addition, results concerning connections of $GE$-groups with abelian groups whose endomorphism rings are unique addition rings have been obtained.