Abstract
A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic p n, for a prime number p, such that its top-factor \(\overline{S} = S/pS\) is a finite semifield. It is well known that if S is an associative Galois Ring (GR) then the set \(S^* = S \ pS\) is a finite multiplicative abelian group. This group is cyclic if and only if S is either a finite field, or a residual integer ring of odd characteristic or the ring ℤ4. A GGR is called top-associative if \(\overline{S}\) is a finite field. In this paper we study the conditions for a top-associative not associative GGR S to be cyclic.
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González, S., Markov, V.T., Martínez, C., Nechaev, A.A., Rúa, I.F. (2004). On Cyclic Top-Associative Generalized Galois Rings. In: Mullen, G.L., Poli, A., Stichtenoth, H. (eds) Finite Fields and Applications. Fq 2003. Lecture Notes in Computer Science, vol 2948. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24633-6_3
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