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 $S = p\Z_{p^t}$ for $p$ prime and $t\ge 3$.
Keywords: circle monoid; circle submonoids of rings; representations and antirepresentations
Classification (MSC2000): 20M25; 20M30
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