Adjoint group structures for cyclic group:Z4

This article gives specific information, namely, adjoint group structures, about a particular group, namely: cyclic group:Z4.
View adjoint group structures for particular groups | View other specific information about cyclic group:Z4

There exist two ways of making cyclic group:Z4 into the adjoint group of a radical ring. One of these makes it an algebra group. The other simply uses a ring with the same additive group.

Summary

Radical ring	Additive group	Simplest commutative unital rings over which it is an algebra	Maximum of element nilpotencies	Nilpotency index of whole ring
nilpotent over field:F2, where one basis element squares to the other and all basis products are zero (see algebra group structures for cyclic group:Z4)	Klein four-group	field:F2	3	3
has additive group cyclic group:Z4, multiplication is identically zero	cyclic group:Z4	ring:Z4	2	2