A group \(G\) is called \(R^*\)-group if for all \(n>0\) and elements \(g\) and \(x_1,\dots,x_n\) the equation \(g^{x_1 }\cdots g^{x_n }=1\) implies \(g=1\). The following results are proved: (1) if \(G\) is an Abelian-by-nilpotent as well as nilpotent-by-Abelian \(R^*\)-group, then every partial order on \(G\) can be extented to a linear order; (2) if \(p(x)\in\mathbb{Q}[x]\) is an irreducible polynomial none of whose roots is a positive real number then there exists a non-zero polynomial \(f(x)\in\mathbb{Q}[x]\) such that all coefficients of \(f(x)\) are positive and \(p(x)\) divides \(f(x)\). This cannot happen if at least one of the roots of \(p(x)\) is a positive real number.