Adjoint group structures for cyclic group:Z8

Below, we discuss various ways of realizing cyclic group:Z8 as the adjoint group of a radical ring. Note that since Z8 is not an algebra group, it is not possible to choose a radical ring whose additive group has exponent two. We can, however, choose a radical ring whose additive group has exponent eight.

Exponent eight
Here the additive group of the radical ring is cyclic group:Z8 and the multiplication is identically the zero map.