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**\(A\)-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.

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.

Reviewer: Ladislav Bican (Praha)

### 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 |

### Keywords:

projective groups; \(\mathcal A\)-decomposable groups; \(A\)-generated groups; \(\mathcal G_{\mathcal A}\)-presented groups; direct sums; direct summands; direct products; pure subgroups### References:

[1] | U. Albrecht: Endomorphism rings of faithfully flat abelian groups of infinite rank. Results Math. 17 (1990), 179-201. · Zbl 0709.20031 |

[2] | U. Albrecht: On the construction of \(A\)-solvable abelian groups. Czechoslovak Math. J. 44 (1994), 413-430. · Zbl 0823.20056 |

[3] | U. Albrecht, H. P. Goeters and W. Wickless: The flat dimension of mixed abelian groups as \(E\)-modules. Rocky Mountain J. Math. 25 (1995), 569-590. · Zbl 0843.20045 |

[4] | F. Anderson and K. Fuller: Rings and Categories of Modules. Springer Verlag, 1992. · Zbl 0765.16001 |

[5] | D. Arnold: Abelian groups flat over their endomorphism ring. Preprint. |

[6] | D. Arnold, R. Hunter and F. Richman: Global Azumaya theorems in additive categories. J. Pure Appl. Algebra 16 (1980), 232-242. · Zbl 0443.18014 |

[7] | D. Arnold and L. Lady: Endomorphism rings and direct sums of torsion-free abelian groups. Trans. Amer. Math. Soc. 211 (1975), 225-237. · Zbl 0329.20033 |

[8] | R. Baer: Abelian groups without elements of finite order. Duke Math. J. 3 (1937), 68-122. · Zbl 0016.20303 |

[9] | M. Dugas and R. Goebel: Every cotorsion-free ring is an endomorphism ring. Proc. London Math. Soc. 45 (1982), 319-336. · Zbl 0506.16022 |

[10] | L. Fuchs: Infinite Abelian Groups. Academic Press, New York, London, 1970/73. · Zbl 0209.05503 |

[11] | S. Glaz and W. Wickless: Regular and principal projective endomorphism rings of mixed abelian groups. · Zbl 0801.20037 |

[12] | P. Griffith: Infinite Abelian Groups. Chicago Lectures in Mathematics, 1970. · Zbl 0204.35001 |

[13] | R. Hunter, F. Richman and E. Walker: Warfield modules. LNM 616, Springer, 1977, pp. 87-139. · Zbl 0376.13007 |

[14] | I. Kaplansky: Projective modules. Ann. of Math. 68 (1958), 372-377. · Zbl 0083.25802 |

[15] | L. Kulikov: On direct decompositions of groups. Ukrain. Mat. Zh. 4 (1952), 230-275, 347-372. |

[16] | B. Stenström: Ring of Quotients. Springer Verlag, Berlin, Heidelberg, New York, 1975. · Zbl 0296.16001 |

[17] | F. Ulmer: A flatness criterion in Grothendick categories. Invent. Math. 19 (1973), 331-336. · Zbl 0257.18020 |

[18] | W. Wickless: A functor from mixed groups to torsion-free groups. Contemp. Math. 171 (1994), 407-417. · Zbl 0821.20036 |

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