Let <M>R</M> be an associative ring, not necessarily with one.
The set of all elements of <M>R</M> forms a monoid with the neutral element
<M>0</M> from <M>R</M> under the operation <M> r \cdot s = r + s + rs </M>
defined for all <M>r</M> and <M>s</M> of <M>R</M>. This operation is called
the <E>circle multiplication</E>, and it is also known as the
<E>star multiplication</E>. The monoid of elements of <M>R</M> under the
circle multiplication is called the adjoint semigroup of <M>R</M> and is
denoted by <M>R^{ad}</M>. The group of all invertible elements of this
monoid is called the adjoint group of <M>R</M> and is denoted by <M>R^{*}</M>.
<P/>
These notions naturally lead to a number of questions about the connection
between a ring and its adjoint group, for example, how the ring properties
will determine properties of the adjoint group; which groups can appear as
adjoint groups of rings; which rings can have adjoint groups with
prescribed properties, etc.
<P/>
For example, V. O. Gorlov in <Cite Key="Gorlov-1995" /> gives
a full list of finite nilpotent algebras <M>R</M>, such that
<M>R^2 \ne 0</M> and the adjoint group of <M>R</M> is
metacyclic (but not cyclic).
<P/>
S. V. Popovich and Ya. P. Sysak in <Cite Key="Popovich-Sysak-1997" />
characterize all quasiregular algebras such that all subgroups
of their adjoint group are their subalgebras. In particular,
they show that all algebras of such type are nilpotent with
nilpotency index at most three.
<P/>
Various connections between properties of a ring and its
adjoint group were considered by O. D. Artemovych and
Yu. B. Ishchuk in <Cite Key="Artemovych-Ishchuk-1997" />.
<P/>
B. Amberg and L. S. Kazarin in <Cite Key="Amberg-Kazarin-2000" />
give the description of all nonisomorphic finite <M>p</M>-groups
that can occur as the adjoint group of some nilpotent
<M>p</M>-algebra of the dimension at most 5.
<P/>
In <Cite Key="Amberg-Sysak-2001" /> B. Amberg and Ya. P. Sysak
give a survey of results on adjoint groups of radical rings,
including such topics as subgroups of the adjoint group; nilpotent
groups which are isomorphic to the adjoint group of some radical
ring; adjoint groups of finite nilpotent $p$-algebras.
The authors continued their investigations in further papers
<Cite Key="Amberg-Sysak-2002" /> and <Cite Key="Amberg-Sysak-2004" />.
<P/>
In <Cite Key="Kazarin-Soules-2004" /> L. S. Kazarin and P. Soules
study associative nilpotent algebras over a field of positive
characteristic whose adjoint group has a small number of generators.
<P/>
The main objective of the proposed &GAP;4 package &Circle; is to
extend the &GAP; functionality for computations in adjoint
groups of associative rings to make it possible to use the &GAP;
system for the investigation of the above described questions.
<P/>
&Circle; provides functionality to construct circle objects that
will respect the circle multiplication <M> r \cdot s = r + s + rs </M>,
create multiplicative structures, generated by such objects,
and compute adjoint semigroups and adjoint groups of finite rings.
<P/>
Also we hope that the package will be useful as an example of
extending the &GAP; system with new multiplicative objects.
Relevant details are explained in the next chapter of the manual.
<Section Label="IntroInstall">
<Heading>Installation and system requirements</Heading>
&Circle; does not use external binaries and, therefore, works without
restrictions on the type of the operating system. This version of the
package is designed for &GAP;4.5 and no compatibility with previous
releases of &GAP;4 is guaranteed.
<P/>
To use the &Circle; online help it is necessary to install the &GAP;4 package
&GAPDoc; by Frank Lübeck and Max Neunhöffer, which is available from the
&GAP; site or from <URL>https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/</URL>.
<P/>
&Circle; is distributed in standard formats
(<File>tar.gz</File>, <File>tar.bz2</File>, <File>zip</File> and <File>-win.zip</File>)
and can be obtained from <URL>https://gap-packages.github.io/circle</URL>
or from the &GAP; homepage.
To install the package, unpack its archive in the <File>pkg</File> subdirectory of your
&GAP; installation.
</Section>
</Chapter>
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