Cyclic Generalized Galois Rings

ABSTRACT

The role played by fields in relation to Galois Rings corresponds to semifields if the associativity is dropped, that is, if we consider Generalized Galois Rings instead of (associative) Galois rings. If S is a Galois ring and pS is the set of zero divisors in S, S* = S\ pS is known to be a finite {multiplicative} Abelian group that is cyclic if, and only if, S is a finite field, or S = ℤ/nℤ with n = 4 or n = p ^r for some odd prime p. Without associativity, S* is not a group, but a loop. The question of when this loop can be generated by a single element is addressed in this article.

Key Words:

• Cyclic condition
• Finite semifield
• Generalized Galois Ring
• Multiplicative structure

Mathematics Subject Classification:

• Primary 17A60, 17A99
• Secondary 17D99

ACKNOWLEDGMENT

Partially supported by MEC04-MTM2004-08115-C04-01.

Communicated by I. Shestakov.