1<Chapter Label="Intro"> 2<Heading>Introduction</Heading> 3 4<Section Label="IntroAbstract"> 5<Heading>General aims</Heading> 6 7Let <M>R</M> be an associative ring, not necessarily with one. 8The set of all elements of <M>R</M> forms a monoid with the neutral element 9<M>0</M> from <M>R</M> under the operation <M> r \cdot s = r + s + rs </M> 10defined for all <M>r</M> and <M>s</M> of <M>R</M>. This operation is called 11the <E>circle multiplication</E>, and it is also known as the 12<E>star multiplication</E>. The monoid of elements of <M>R</M> under the 13circle multiplication is called the adjoint semigroup of <M>R</M> and is 14denoted by <M>R^{ad}</M>. The group of all invertible elements of this 15monoid is called the adjoint group of <M>R</M> and is denoted by <M>R^{*}</M>. 16<P/> 17 18These notions naturally lead to a number of questions about the connection 19between a ring and its adjoint group, for example, how the ring properties 20will determine properties of the adjoint group; which groups can appear as 21adjoint groups of rings; which rings can have adjoint groups with 22prescribed properties, etc. 23<P/> 24 25For example, V. O. Gorlov in <Cite Key="Gorlov-1995" /> gives 26a full list of finite nilpotent algebras <M>R</M>, such that 27<M>R^2 \ne 0</M> and the adjoint group of <M>R</M> is 28metacyclic (but not cyclic). 29<P/> 30 31S. V. Popovich and Ya. P. Sysak in <Cite Key="Popovich-Sysak-1997" /> 32characterize all quasiregular algebras such that all subgroups 33of their adjoint group are their subalgebras. In particular, 34they show that all algebras of such type are nilpotent with 35nilpotency index at most three. 36<P/> 37 38Various connections between properties of a ring and its 39adjoint group were considered by O. D. Artemovych and 40Yu. B. Ishchuk in <Cite Key="Artemovych-Ishchuk-1997" />. 41<P/> 42 43B. Amberg and L. S. Kazarin in <Cite Key="Amberg-Kazarin-2000" /> 44give the description of all nonisomorphic finite <M>p</M>-groups 45that can occur as the adjoint group of some nilpotent 46<M>p</M>-algebra of the dimension at most 5. 47<P/> 48 49In <Cite Key="Amberg-Sysak-2001" /> B. Amberg and Ya. P. Sysak 50give a survey of results on adjoint groups of radical rings, 51including such topics as subgroups of the adjoint group; nilpotent 52groups which are isomorphic to the adjoint group of some radical 53ring; adjoint groups of finite nilpotent $p$-algebras. 54The authors continued their investigations in further papers 55<Cite Key="Amberg-Sysak-2002" /> and <Cite Key="Amberg-Sysak-2004" />. 56<P/> 57 58In <Cite Key="Kazarin-Soules-2004" /> L. S. Kazarin and P. Soules 59study associative nilpotent algebras over a field of positive 60characteristic whose adjoint group has a small number of generators. 61<P/> 62 63The main objective of the proposed &GAP;4 package &Circle; is to 64extend the &GAP; functionality for computations in adjoint 65groups of associative rings to make it possible to use the &GAP; 66system for the investigation of the above described questions. 67<P/> 68 69&Circle; provides functionality to construct circle objects that 70will respect the circle multiplication <M> r \cdot s = r + s + rs </M>, 71create multiplicative structures, generated by such objects, 72and compute adjoint semigroups and adjoint groups of finite rings. 73<P/> 74 75Also we hope that the package will be useful as an example of 76extending the &GAP; system with new multiplicative objects. 77Relevant details are explained in the next chapter of the manual. 78 79</Section> 80 81<!-- ********************************************************* --> 82 83<Section Label="IntroInstall"> 84<Heading>Installation and system requirements</Heading> 85 86&Circle; does not use external binaries and, therefore, works without 87restrictions on the type of the operating system. This version of the 88package is designed for &GAP;4.5 and no compatibility with previous 89releases of &GAP;4 is guaranteed. 90<P/> 91 92To use the &Circle; online help it is necessary to install the &GAP;4 package 93&GAPDoc; by Frank Lübeck and Max Neunhöffer, which is available from the 94&GAP; site or from <URL>http://www.math.rwth-aachen.de/˜Frank.Luebeck/GAPDoc/</URL>. 95<P/> 96 97&Circle; is distributed in standard formats 98(<File>tar.gz</File>, <File>tar.bz2</File>, <File>zip</File> and <File>-win.zip</File>) 99and can be obtained from <URL>http://www.cs.st-andrews.ac.uk/˜alexk/circle/</URL> 100or from the &GAP; homepage. 101To install the package, unpack its archive in the <File>pkg</File> subdirectory of your 102&GAP; installation. 103</Section> 104 105</Chapter>