On the projective commutant of Abelian groups.
In this work the investigations of the author's paper [``On projective invariant subgroups of Abelian groups'', J. Math. Sci., New York 164, No. 1, 143-147 (2010); translation from Fundam. Prikl. Mat. 14, No. 6, 211-218 (2008)] are continued. The author uses the following concepts and notations. By \(M\) he denotes some module and by \(E(M)\) its endomorphism ring; \(\text{Pr}(M)=\text{Id}(E(M))\). If \(H\leq M\) and \(\pi H\subseteq H\) for all \(\pi\in\text{Pr}(M)\) then the submodule \(H\) is called projectively invariant (briefly, a pi-submodule) and denoted by \(H\leq\text{pi\,}M\). Call a submodule \(H\leq M\) commutator projectively invariant (briefly, a cpi-submodule) if \([\xi,\eta]H\subseteq H\) for all \(\xi,\eta\in\text{Pr}(M)\). Clearly, if \(H\leq\text{pi\,}M\), then \(H\leq\text{cpi\,}M\). By the projective center (briefly, the P-center) of \(M\) he means the submodule: \(PZ(M)=\{a\in M\mid[\xi,\eta]a=0\text{ for all }\xi,\eta\in\text{Pr}(M)\}\). Call the submodule \(\text{P}(M)=\langle[\varphi,\psi]M\mid\varphi,\psi\in\text{Pr}(M)\rangle\) the projective commutant (briefly, the P-commutant) of \(M\). The main results of the paper are the following: Theorem 1. Suppose that \(A=B\oplus D\), where \(D=t(D)\oplus D_0\) is the divisible part of \(A\) and \(H\leq A\). Then \(H\leq\text{cpi\,}A\) if and only if \(H\) coincides with one of the following subgroups: (1) \(H=F\oplus(\bigoplus_pD_p[p^{k_p}])\), where \(F\) is a periodic cpi-subgroup in \(B\) and \(k_p\geq\sup\{e(b)\mid b\in F_p\}\); (2) \(H=G\oplus(\bigoplus_{p\in\Pi_1}D_p[p^{k_p}])\), where \(G\) is aperiodic cpi-subgroup in \(B\oplus(\bigoplus_{p\in\Pi}D_p)\) as in Lemma 4(2) \(k_p\geq\sup\{e(g)\mid g\in G_p\}\) and \(\Pi_1\cap\Pi=\emptyset\); (3) \(H=C\oplus D\), where \(C\leq\text{cpi\,}B\); (4) \(H=E\oplus t(D)\), \(r(D_0)=1\) and \(E\) is a nonperiodic cpi-subgroup in \(B\oplus D_0\) as in Lemma 4(1). Theorem 2. In a decomposable reduced separable torsion-free group \(A\), all cpi-subgroups are fully invariant if and only if, for every direct summand \(B\) of rank 1 of the group \(A\), in the complementary direct summand there exists a direct summand \(G\) isomorphic to \(B\). Theorem 3. In a reduced algebraically compact torsion-free group \(A\), its every cpi-subgroup is fully invariant if and only if all \(p\)-adic components of \(A\) are decomposable. Theorem 5. Suppose that \(A=B\oplus D\), where \(D=t(D)\oplus D_0\) is the nonzero divisible part of \(A\), and suppose that \(G=PZ(B)\cap(\bigcap_{\varphi\in\Hom(B,D)}\ker\varphi)\). Then \(G\) is a periodic subgroup, \(G=\bigoplus_{p\in\Pi}G_p\) (\(G_p\neq 0\) for \(p\in\Pi\)), \(D_p=0\) for each \(p\in\Pi\), and \(PZ(A)\) coincides with one of the following subgroups: (1) if \(t(D)\neq 0\) then \(PZ(A)=G\oplus(\bigoplus_{p\in\Pi_1}D_p)\), where \(\Pi_1=\{p\in P\mid r(D_p)=1\}\); (2) if \(t(D)=0\) then either \(PZ(A)=G\) or, if \(r(D_0)=1\), then \(PZ(A)=G\oplus D_0\).
reduced separable torsion-free Abelian groups
scientific article; zbMATH DE number 6105687