ABSOLUTE IDEALS OF ALMOST COMPLETELY DECOMPOSABLE ABELIAN GROUPS

A ring is said to be a ring on an abelian group G, if its additive group coincides with the group G. A subgroup of the group G is called the absolute ideal of G, if it is an ideal of every ring on the group G. If every ideal of a ring is an absolute ideal of its additive group, then the ring is called the AI-ring. If there exists at least one AI-ring on a group G, then the group G is called the RAI-group. We consider rings on almost completely decomposable abealian groups (acd-groups) in the present paper. A torsion free abelian group is an acd-group, if it contains a completely decomposable subgroup of finite rank and of finite index. Every acd-group G contains the regulator A, which is completely decomposable and fully invariant. The finite quotient group G/A is called the regulator quotient of the group G, the order of the group G/A is called the regulator index. If the regulator quotient of an acd-group is cyclic, then the group is called the crq-group. If the types of the direct rank-1 summands of the regulator A are pairwise incomparable, then the groups A and G are called rigid. If all these types are idempotent, then the group G is of the ring type. The main result of the present paper is that every rigid crq-group of the ring type is an RAI-group. Moreover, the principal absolute ideals are completely described for such groups. Let G be a rigid crq-group of the ring type. A subgroup A is the regulator of the group G, the quotient G/A = ⟨d + A⟩ is the regulator quotient and n is the regulator index. A decomposition A = ⊕ τ∈T(G) Aτ of the regulator A into a direct sum of rank-1groups Aτ determines the set T(G) = T(A) of critical types of the groups A and G. Then for every τ ∈ T(G), there exists an element eτ ∈ Aτ such that A = ⊕ τ∈T(G) Rτ eτ , where Rτ (τ ∈ T(G)) is a subring of the field of rational numbers containing the unit. Moreover, the definition of natural near-isomorphism invariants mτ (τ ∈ ∈ T(G)) of the group G naturally implies that every element g ∈ G can be written in the divisible hull of the group G in the following way g = ∑ τ∈T(G) rτ mτ eτ , where rτ are elements of the ring Rτ which are uniquely determined by a fixed decomposition of the regulator A. Every description of RAI-groups is based on a description of principal absolute ideals of the groups. The least absolute ideal ⟨g⟩AI containing an element g is called the principal absolute ideal generating by g. The following theorem describes principal absolute ideals. Theorem 1. Let G be a rigid crq-group of the ring type with a fixed decomposition of the regulator, g = ∑ τ∈T(G) rτ mτ eτ ∈ G. Then ⟨g⟩AI = ⟨g⟩ + ⊕ τ∈T(G) rτAτ . Note that the elements rτ (τ ∈ T(G)) in the representation of the element g ∈ G are determined uniquely up to an invertible factor of Rτ . Therefore, the representation of the principal absolute ideal doesn’t depend on the decomposition of the regulator. Theorem 2. Every rigid crq-group G of the ring type is an RAI-group. In this case, for every integer α соprime to n there exists an AI-ring (G, ×) such that the equality d × d = αd takes place in the quotient ring (G/A, ×), where d = d + A, G/A = ⟨d⟩.

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