Abelian SACR-groups
A homomorphism μ: G ⊗ G → G is called a multiplication on an abelian group G. An abelian group G with a multiplication on it is called a ring on G. The study of abelian groups supporting only a certain ring is one of the trends in the additive group theory. An abelian group on which every ring is associative and commutative is called an SACR-group (this abbreviation comes from: “strongly associative and commutative ring”). In this paper, we study SACR-groups in the following classes of abelian groups: homogeneous completely decomposable quotient divisible groups and indecomposable torsion-free groups of rank 2. Together with associative and commutative rings, we are also interested in additive groups of filial rings. An associative ring in which all meta-ideals of finite index are ideals is called filial. Certainly, an associative ring R is called filial if the relation of being an ideal in R is transitive. An abelian group on which every associative ring is filial is called a TI-group. In Section 1, homogeneous completely decomposable quotient divisible abelian SACR-groups are described (Theorem 7). The proof of this theorem is based on Theorem 4: every quotient divisible group of rank 1 is an SACR-group. Further, in Section 3, it is shown that every indecomposable torsion-free group of rank 2 is an SACR-group. In particular, TI-groups are described in the class of indecomposable torsion-free abelian groups of rank 2. It is shown that the concepts of a TI-group and a nil-group in the class of rank 2 torsion-free indecomposable groups are equivalent. Until now, all known torsion-free TI-groups are SACR-groups. However, the converse is not true; an example is given in Section 3. AMS Mathematical Subject Classification: 20K15, 20K21
Keywords
абелева группа,
кольцо на группе,
SACR-группы,
TI-группы,
abelian group,
ring on group,
SACR-group,
TI-groupAuthors
| Nguyen Thi Quynh Trang | Moscow Pedagogical State University | trangnguyen.ru@gmail.com |
Всего: 1
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