## $$A$$-projective resolutions and an Azumaya theorem for a class of mixed Abelian groups.(English)Zbl 1079.20503

If $$A$$ is an Abelian group then any direct summand of a direct power of $$A$$ is called $$A$$-projective. If $$\mathcal A$$ is a class of groups then an Abelian group $$G$$ is said to be $$\mathcal A$$-decomposable if it is of the form $$G\cong\bigoplus_{A\in \mathcal A}P_A$$ where each $$P_A$$ is $$A$$-projective. Let $$\Gamma$$ denote the class of groups $$G$$ which are isomorphic to pure subgroups of $$\prod_pG_p$$ containing $$\bigoplus_pG_p$$. The subclass of $$\Gamma$$ consisting of all groups having finite torsionfree rank is denoted by $$\Gamma_\infty$$. Now every $$G\in\Gamma_\infty$$ contains a finite independent subset $$X$$ such that $$F=\langle X\rangle$$ is a free subgroup of $$G$$ with $$G/F$$ torsion. Viewing $$G$$ as a pure subgroup of $$\prod_pG_p$$ we can write $$X=\{x_i=(x_{ip})\mid i=1,\dots,n\}$$. Finally the class $$\mathcal G$$ consists of all elements of $$\Gamma_\infty$$ for which $$G_p$$ is finite for all $$p$$ and satisfies $$G_p=\langle x_{1p},\dots,x_{np}\rangle$$ for all but finitely many $$p$$.
The main purpose of the paper is to prove an Azumaya-style theorem stating that the class of $$\mathcal G$$-decomposable groups is closed under direct summands.

### MSC:

 20K21 Mixed groups 20K25 Direct sums, direct products, etc. for abelian groups 20K27 Subgroups of abelian groups 20K40 Homological and categorical methods for abelian groups
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### References:

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