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We classify the quadratic extensions K = Q[ √ d] and the finite groups G for which the group ring oK [G] of G over the ring oK of integers of K has the property that the group U1(oK [G]) of units of augmentation 1 is hyperbolic. We also construct units in the Z-order H(oK) of the quaternion algebra H(K) = ` −1, −1 K ´ , when it is a division algebra.

- S O Juriaans, C Polcino, Millies, A C Souza Filho
- 2009

We prove a structure theorem for the alternative finite dimensional algebras over a field K, which can be the racional numbers or an imaginary racional quadratic extension, with the hyperbolic property. One class of such algebras is the alternative totally definite octonion algebra over K. We classify the RA-loops L for which the unit loop of its integral… (More)

- S. O. Juriaans
- 2009

In [4] it was shown for a suitable power n of a pair of units u, v of the quaternions algebras over the ring of integers of imaginary rational extensions A = H(o Q √ −d) that the group generated by u n , v n is a free group in the unit group of A when d ≡ 7 (mod 8) is a positive square free integer. We extend this result for any imaginary rational extension… (More)

In 1996 Jespers and Wang classified finite semigroups whose integral semigroup ring has finitely many units. In a recent paper, Iwaki-Juriaans-Souza Filho continued this line of research by partially classifying the finite semigroups whose rational semigroup algebra contains a Z-order with hyperbolic unit group. In this paper we complete this classification… (More)

For a given division algebra of the quaternions, we construct two types of units of its Z-orders: Pell units and Gauss units. Also, if K = Q √ −d, d ∈ Z \ {0, 1} is square free and R = IK, we classify R and G such that U1(RG) is hyperbolic. In particular, we prove that U1(RK8) is hyperbolic iff d > 0 and d ≡ 7 (mod 8). In this case, the hyperbolic boundary… (More)

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