Decompositions of Rings under the Circle Operation


Clare Coleman and David Easdown


Research Report 99-19
Date: 31 August 1999


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


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