chap1.txt

  
  1 Introduction
  
  
  1.1 General aims
  
  Let  R  be  an  associative  ring,  not necessarily with one. The set of all
  elements  of  R  forms  a monoid with the neutral element 0 from R under the
  operation r ⋅ s = r + s + rs defined for all r and s of R. This operation is
  called  the  circle  multiplication,  and  it  is  also  known  as  the star
  multiplication.  The monoid of elements of R under the circle multiplication
  is  called  the  adjoint semigroup of R and is denoted by R^ad. The group of
  all  invertible elements of this monoid is called the adjoint group of R and
  is denoted by R^*.
  
  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.
  
  For  example,  V. O. Gorlov in [Gor95] gives a full list of finite nilpotent
  algebras R, such that R^2 ne 0 and the adjoint group of R is metacyclic (but
  not cyclic).
  
  S.  V.  Popovich  and  Ya.  P. Sysak in [PS97] 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.
  
  Various  connections between properties of a ring and its adjoint group were
  considered by O. D. Artemovych and Yu. B. Ishchuk in [AI97].
  
  B.  Amberg  and  L.  S.  Kazarin  in  [AK00]  give  the  description  of all
  nonisomorphic  finite  p-groups  that can occur as the adjoint group of some
  nilpotent p-algebra of the dimension at most 5.
  
  In  [AS01]  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 [AS02] and [AS04].
  
  In  [KS04]  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.
  
  The  main objective of the proposed GAP4 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.
  
  Circle  provides functionality to construct circle objects that will respect
  the  circle  multiplication  r  ⋅  s  =  r  +  s + rs, create multiplicative
  structures,  generated  by  such objects, and compute adjoint semigroups and
  adjoint groups of finite rings.
  
  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.
  
  
  1.2 Installation and system requirements
  
  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 GAP4.5 and no compatibility with previous releases
  of GAP4 is guaranteed.
  
  To  use  the  Circle online help it is necessary to install the GAP4 package
  GAPDoc  by  Frank Lübeck and Max Neunhöffer, which is available from the GAP
  site or from http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/.
  
  Circle  is  distributed  in  standard  formats  (tar.gz,  tar.bz2,  zip  and
  -win.zip)          and          can         be         obtained         from
  http://www.cs.st-andrews.ac.uk/~alexk/circle/  or  from the GAP homepage. To
  install  the package, unpack its archive in the pkg subdirectory of your GAP
  installation.