On endoprimality of torsion-free Abelian groups of rank 3.
Let \(A\) be a left \(R\)-module and \(n\in\mathbb{N}^*\). A function \(f\colon A^n\to A\) is an endofunction of \(A\) if \(\varphi(f(a_1,\dots,a_n))=f(\varphi(a_1),\dots,\varphi(a_n))\) for every endomorphism \(\varphi\) of \(A\) and for all \(a_1,\dots,a_n\in A\). We say that \(A\) is endoprimal if every endofunction of \(A\) is a term function which means that for every endofunction \(f\colon A^n\to A\) there exist \(r_1,\dots,r_n\in R\) such that \(f(a_1,\dots,a_n)=r_1a_1+\cdots+r_na_n\). If \(F\) is a field and \(R\) is an \(F\)-algebra a function \(f\colon A^n\to A\) is called an \(R\)-function if it is homogeneous, which means that \(f(ra_1,\dots,ra_n)=rf(a_1,\dots,a_n)\) for all \(a_1,\dots,a_n\in A\). \(A\) is called a rich module if every \(R\)-function of \(A\) is a term function of \(A\) as an \(F\)-linear space. In the present paper, which is a natural continuation of [\textit{K. Kaarli} and \textit{L. Márki}, J. Aust. Math. Soc., Ser. A 67, No. 3, 412-428 (1999; Zbl 0944.20037)], the authors determine all rank 3 torsion free groups which are endoprimal as modules over their nucleus (Theorem 1.1). This result is obtained as a consequence of some general result about rich modules over unital \(F\)-algebras, where \(F\) is a field (Theorem 3.5 and Theorem 4.7).
scientific article; zbMATH DE number 2186350