Twisted Representations of Semiadditive Rings (Dedicated to Professor D.K. Ray-Chaudhuri) NAVIN SINGHI Abstract Semiadditive rings were defined by the author in a recent paper. The theory of semiadditive rings is being developed to create an algebraic tool to study the finite projective planes. The recent results proved on semiadditive rings are described. A new twisted representation is created for semiadditive rings. The twisting additive subgroup for this representation is an ideal in a polynomial ring over the ring of integers. 1. INTRODUCTION A semiadditive ring with a multiplicative identity is an additive loop (R, +) together with a ternary operation T on R such that for all x, y ∈ R, T(x, 0, y) = T(0, x, y) = y, T(1, x, 0) = T(x, 1, 0) = x and T(1, x, y) = x + y (see section 2 for more detailed definitions and results). Semiadditive rings were defined and studied by the author in [S]. The theory of semiadditive rings is being developed to create a tool to study the finite projective planes. Two well known conjectures on projective planes state that the order of a finite projective plane is a power of a prime number and a finite projective plane with no proper subplane is a Desarguesian projective plane over a finite prime field Z p . For more details on relationships of semiadditive rings with projective planes and many other related structures see [S]. For some interesting results and other algebraic methods, in recent years on this topic see [A, M]. In the next section we will describe without proofs main results proved on semiadditive rings and free semiadditive rings. The ring of integers Z or a polynomial ring over Z may be thought of as a free semiadditive ring satisfying additional conditions like associativity, commutativity, distributivity or linearity (see Section 2 for more details). Thus a free semiadditive ring is an analogue of the ring of integers when these conditions are not satisfied. Using a J. of Combinatorics, Information & System Sciences Vol. 34 (2009) No. 2, pages 241-254 © MD Publications Pvt Ltd Corresponding author email: singhi@math.tifr.res.in