On the multiplicative structure of finite division rings (Q1175449)

A finite division ring is a finite algebraic system satisfying all the field axioms except commutativity and associativity of multiplication. The element \(p\) is right primitive if it generates all nonzero elements by right multiplication with the identity. The author investigates the conjecture that every finite division ring contains a right primitive element. It is shown that all division rings with 16, 27, 32, 125, or 343 elements and all commutative division rings three-dimensional over a finite field not of characteristic two have a right primitive element. Examples of division rings of order \(2^ 7\), \(2^ 9\), and \(2^{11}\) with right primitive elements are given. Another example shows that right primitive elements need not occur if one assumes only the existence of a right identity.