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**Endomorphism rings of faithfully flat Abelian groups.**
*(English)*
Zbl 0709.20031

An abelian group A is called faithfully flat if A is flat when considered as a left module over its endomorphism ring End(A) and IA\(\neq A\) for all proper right ideals I of End(A). At least this is the definition given in the preprint “Abelian groups flat over their endomorphism rings” by D. Arnold (reference [Ar1]). I can only assume it is the definition the author is using, since he gives no direct evidence on the matter. The reader is referred to the paper for undefined terms used below. The main thrust of the paper is to study relationships between a faithfully flat abelian group A and certain classes defined in terms of A. For example a class \({\mathcal T}\) of abelian groups is called A-balanced closed provided the following five conditions are satisfied.

(1) For each \(G\in {\mathcal T}\), \(G=S_ A(G)=\sum \{\phi (A)|\phi\in Hom(A,G)\}.\)

(2) \({\mathcal T}\) is closed under finite direct sums.

(3) If U is a subgroup of \(G\in {\mathcal T}\) with \(S_ A(U)=U\), then \(U\in {\mathcal T}.\)

(4) If B,C\(\in {\mathcal T}\) and \(\phi\in Hom(B,C)\), then ker \(\phi\in {\mathcal T}.\)

(5) A is projective with respect to any exact sequence of elements of \({\mathcal T}.\)

Theorem 2.5. The following are equivalent for a self-small abelian group.

(a) A is faithfully flat.

(b) There exists an A-balaced closed class \({\mathcal T}\) which contains the A-projective groups.

(c) The class of A-solvable groups is the largest A-balanced closed class which contains A.

The author shows that if R is a countable reduced torsion-free ring with identity, then the Corner construction of a group A with End(A)\(\simeq R\) produces a self-small faithfully flat A. Moreover, the Black Box methods of Corner-Göbel can be modified to produce a self-small faithfully flat A with prescribed endomorphism ring.

In the third section of the paper, the author obtains several stronger versions of Theorem 2.5 by imposing additional conditions on End(A). For example, a result proved in a previous paper by the author is that if A is self-small, then A is flat as an End(A)-module and End(A) is right hereditary if and only if the class of A-projectives is A-balanced closed.

In the fourth section, the author studies the quasi-splitting of exact sequences involving groups G with \(S_ A(G)=G\).

(1) For each \(G\in {\mathcal T}\), \(G=S_ A(G)=\sum \{\phi (A)|\phi\in Hom(A,G)\}.\)

(2) \({\mathcal T}\) is closed under finite direct sums.

(3) If U is a subgroup of \(G\in {\mathcal T}\) with \(S_ A(U)=U\), then \(U\in {\mathcal T}.\)

(4) If B,C\(\in {\mathcal T}\) and \(\phi\in Hom(B,C)\), then ker \(\phi\in {\mathcal T}.\)

(5) A is projective with respect to any exact sequence of elements of \({\mathcal T}.\)

Theorem 2.5. The following are equivalent for a self-small abelian group.

(a) A is faithfully flat.

(b) There exists an A-balaced closed class \({\mathcal T}\) which contains the A-projective groups.

(c) The class of A-solvable groups is the largest A-balanced closed class which contains A.

The author shows that if R is a countable reduced torsion-free ring with identity, then the Corner construction of a group A with End(A)\(\simeq R\) produces a self-small faithfully flat A. Moreover, the Black Box methods of Corner-Göbel can be modified to produce a self-small faithfully flat A with prescribed endomorphism ring.

In the third section of the paper, the author obtains several stronger versions of Theorem 2.5 by imposing additional conditions on End(A). For example, a result proved in a previous paper by the author is that if A is self-small, then A is flat as an End(A)-module and End(A) is right hereditary if and only if the class of A-projectives is A-balanced closed.

In the fourth section, the author studies the quasi-splitting of exact sequences involving groups G with \(S_ A(G)=G\).

Reviewer: C.Vinsonhaler

### MSC:

20K20 | Torsion-free groups, infinite rank |

20K30 | Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups |

16S50 | Endomorphism rings; matrix rings |

16D40 | Free, projective, and flat modules and ideals in associative algebras |

20K25 | Direct sums, direct products, etc. for abelian groups |

### Keywords:

endomorphism ring; faithfully flat abelian group; self-small abelian group; A-balaced closed class; A-projective groups; A-solvable groups; countable reduced torsion-free ring; Corner construction; Black Box; quasi-splitting; exact sequences
Full Text:
DOI

### References:

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