<p>Let <span class="SimpleMath">\(R\)</span> be an associative ring, not necessarily with one. The set of all elements of <span class="SimpleMath">\(R\)</span> forms a monoid with the neutral element <span class="SimpleMath">\(0\)</span> from <span class="SimpleMath">\(R\)</span> under the operation <span class="SimpleMath">\( r \cdot s = r + s + rs \)</span> defined for all <span class="SimpleMath">\(r\)</span> and <span class="SimpleMath">\(s\)</span> of <span class="SimpleMath">\(R\)</span>. This operation is called the <em>circle multiplication</em>, and it is also known as the <em>star multiplication</em>. The monoid of elements of <span class="SimpleMath">\(R\)</span> under the circle multiplication is called the adjoint semigroup of <span class="SimpleMath">\(R\)</span> and is denoted by <span class="SimpleMath">\(R^{ad}\)</span>. The group of all invertible elements of this monoid is called the adjoint group of <span class="SimpleMath">\(R\)</span> and is denoted by <span class="SimpleMath">\(R^{*}\)</span>.</p>
<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>
<p>For example, V. O. Gorlov in <a href="chapBib_mj.html#biBGorlov-1995">[Gor95]</a> gives a full list of finite nilpotent algebras <span class="SimpleMath">\(R\)</span>, such that <span class="SimpleMath">\(R^2 \ne 0\)</span> and the adjoint group of <span class="SimpleMath">\(R\)</span> is metacyclic (but not cyclic).</p>
<p>S. V. Popovich and Ya. P. Sysak in <a href="chapBib_mj.html#biBPopovich-Sysak-1997">[PS97]</a> 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>
<p>Various connections between properties of a ring and its adjoint group were considered by O. D. Artemovych and Yu. B. Ishchuk in <a href="chapBib_mj.html#biBArtemovych-Ishchuk-1997">[AI97]</a>.</p>
<p>B. Amberg and L. S. Kazarin in <a href="chapBib_mj.html#biBAmberg-Kazarin-2000">[AK00]</a> give the description of all nonisomorphic finite <span class="SimpleMath">\(p\)</span>-groups that can occur as the adjoint group of some nilpotent <span class="SimpleMath">\(p\)</span>-algebra of the dimension at most 5.</p>
<p>In <a href="chapBib_mj.html#biBAmberg-Sysak-2001">[AS01]</a> 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 <a href="chapBib_mj.html#biBAmberg-Sysak-2002">[AS02]</a> and <a href="chapBib_mj.html#biBAmberg-Sysak-2004">[AS04]</a>.</p>
<p>In <a href="chapBib_mj.html#biBKazarin-Soules-2004">[KS04]</a> 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>
<p>The main objective of the proposed <strong class="pkg">GAP</strong>4 package <strong class="pkg">Circle</strong> is to extend the <strong class="pkg">GAP</strong> functionality for computations in adjoint groups of associative rings to make it possible to use the <strong class="pkg">GAP</strong> system for the investigation of the above described questions.</p>
<p><strong class="pkg">Circle</strong> provides functionality to construct circle objects that will respect the circle multiplication <span class="SimpleMath">\( r \cdot s = r + s + rs \)</span>, create multiplicative structures, generated by such objects, and compute adjoint semigroups and adjoint groups of finite rings.</p>
<p>Also we hope that the package will be useful as an example of extending the <strong class="pkg">GAP</strong> system with new multiplicative objects. Relevant details are explained in the next chapter of the manual.</p>
<h4>1.2 <span class="Heading">Installation and system requirements</span></h4>
<p><strong class="pkg">Circle</strong> 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 <strong class="pkg">GAP</strong>4.5 and no compatibility with previous releases of <strong class="pkg">GAP</strong>4 is guaranteed.</p>
<p>To use the <strong class="pkg">Circle</strong> online help it is necessary to install the <strongclass="pkg">GAP</strong>4 package <strong class="pkg">GAPDoc</strong> by Frank Lübeck and Max Neunhöffer, which is available from the <strong class="pkg">GAP</strong> site or from <span class="URL"><a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/">https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/</a></span>.</p>
<p><strong class="pkg">Circle</strong> is distributed in standard formats (<code class="file">tar.gz</code>, <code class="file">tar.bz2</code>, <code class="file">zip</code> and <code class="file">-win.zip</code>) and can be obtained from <span class="URL"><a href="https://gap-packages.github.io/circle">https://gap-packages.github.io/circle</a></span> or from the <strong class="pkg">GAP</strong> homepage. To install the package, unpack its archive in the <code class="file">pkg</code> subdirectory of your <strong class="pkg">GAP</strong> installation.</p>
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