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Zbl 1186.20037
Square subgroups of rank two Abelian groups.
(English)
[J] Colloq. Math. 117, No. 1, 19-28 (2009). ISSN 0010-1354; ISSN 1730-6302/e

The authors deal with the question whether a multiplication $*$ can be defined on a given Abelian group $G$ that makes it into a (not necessarily associative) ring $R=(G,*)$. A group $G$ is nil' if the only way to define a ring structure on $G$ is via the trivial multiplication $x*y=0$. A group $G$ is nil modulo' its subgroup $H$ if for any ring structure $(G,*)$ necessarily $G*G\subseteq H$. The `square subgroup' $\square\,G$ is defined by $\square\,G=\bigcap\{H\subseteq G\mid G\text{ is nil modulo }H\}$.\par Then $G$ is nil if and only if $G$ is nil modulo $\{0\}$ and $\square\,G$ is the smallest subgroup of $G$ such that $G$ is nil modulo $\square\,G$. What is $\square\,G$ and is $G/\square\,G$ nil? It is known that $G/\square\,G$ need not be nil in general. A rank one group $A$ that is non-nil has idempotent type and as a non-trivial ring is isomorphic to a subring of the field of rational numbers $\bbfQ$, hence is a principal ideal domain and $\square\,A=A$.\par The authors tackle the rank two case and obtain two results.\par Theorem~3.4. Let $G$ be a torsion-free group of rank two whose typeset has cardinality $\geq 3$. Then $\square\,G$ is pure in $G$ and $G/\square\,G$ is nil.\par Theorem~4.2. Suppose that $G$ is an indecomposable group of rank two with typeset of cardinality $2$. Then $\square\,G$ is pure in $G$ and $G/\square\,G$ is nil.
MSC 2000:
*20K15 Torsion free abelian groups, finite rank
20K27 Subgroups of abelian groups
16U99 Conditions on elements of noncommutative rings
17A99 General nonassociative rings

Keywords: ring structures; nil rings; additive groups of rings; nil groups; square subgroups; ring multiplications; typesets

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