# Decompositions of Rings under the Circle Operation

## Author

**Clare Coleman** and **David Easdown**

## Status

Research Report 99-19

Date: 31 August 1999

## Abstract

We consider rings S , not necessarily with 1 , and develop a decomposition
theory for submonoids and subgroups of (S, circ) where the circle operation
circ is defined by x circ y = x + y - xy . Decompositions are expressed in
terms of internal semidirect, reverse semidirect and general products, which
may be realised externally in terms of naturally occurring representations
and antirepresentations. The theory is applied to matrix rings over S when
S is radical, obtaining group presentations in terms of
(S, +) and (S, circ). Further details are worked out in special
cases when S = pZ_{p^t} for p prime and t > 2 .

## Key phrases

ring. circle operation. monoid. group. semidirect and general products.
presentations. matrices over a radical ring.

## AMS Subject Classification (1991)

Primary: 16N20

Secondary: 16U60, 15A33, 16W20, 20D40, 20F05, 20H25, 20M10

## Content

The paper is available in the following forms:
- TeX dvi format:
- 1999-19.dvi.gz (48kB) or
1999-19.dvi (133kB)

- PostScript:
- 1999-19.ps.gz (132kB) or
1999-19.ps (367kB)

##### To minimize network load, please choose the smaller gzipped .gz form if
and only if your browser client supports it.

Sydney Mathematics and Statistics