Enumeration of Involutions of Finite Rings

Abstract

In this paper, we study a special class of elements in the finite commutative rings called involutions. An involution of a ring R is an element with the property that x^2-1=0 for some x in R. This study describes both the implementation and enumeration of the involutions of various rings, such as cyclic rings, non-cyclic rings, zero-rings, finite fields, and especially rings of Gaussian integers. The paper begins with simple well-known results of an equation x^2-1=0 over the finite commutative ring R. It provides a concrete setting to enumerate the involutions of the finite cyclic and non-cyclic rings R, along with the isomorphic relation I(R)≅Z_2^k.

Keywords

• cyclic rings
• noncyclic rings
• zero rings
• finite fields
• involutions