Abstract:  A associative ring R with an unit element is called an Involutive Ring,if every element of the muttiplicative group U(R),which consists of all the units of R,satisfies equation z²=e,where e is the unit element of R:In this paper,we have proved that the residue class ring Z.modulo n>1 is an involutive ring if and only if n are the fnllowing numbers:
2,3,4,6,8.12,24. Moreover,we call a ring R is an U-cyclic ring if the unit group U(R)is a cyclic group.  We have proved also that the involutive ring Z,modulon>1is an U-cyclic if and only if n are the following integer  numbers:2.3,1,6.

Key words: unit grouprinvolutive group, involutive ring, cyclic ring, U-cyclic ring