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 \( r \cdot s = r + s + rs \) 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 **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.

**Circle** provides functionality to construct circle objects that will respect the circle multiplication \( r \cdot s = r + s + rs \), 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.

**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.

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 https://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 https://gap-packages.github.io/circle or from the **GAP** homepage. To install the package, unpack its archive in the `pkg`

subdirectory of your **GAP** installation.

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