Please answer to both A and B An element r of a ring R is ca

Please answer to both A and B

An element r of a ring R is called nilpotent if one of its powers is 0, i.e. if there an n belongs to Z_gr 1 such that r^n = 0. (In particular, 0 is always nilpotent by definition. Show that 6 is a nilpotent element of the ring Z/12Z. Let n belongs to greaterthanorequalto 2. Show that Z/nZ has a non-zero nilpotent element if and only if there is a prime p such that p^2 divides n (i.e. n is not squarefree).

Solution

(a) As Z/12Z = (Z/4Z) × (Z/3Z), these factors have coprime order, and the group of units of each of these factors is Z/2Z, it follows that the unit group of Z/2Z is isomorphic to (Z/2Z) 2 ; its elements are multiplication maps by the numbers relatively prime to 12, namely 1, 5, 7, and 11

The nilpotent elements of Z/12Z

where 12 = 2².3¹ are as follows 0,6,12 Hence 6 is a nil potent element of the ring Z/12Z

(b) Let n Z_>2 then Z/nZ will have a non zero nil potent because it is given that Z is greater than or equal to 2 now for nZ relative prime will be 5,7,11,....

Now p²=25,49,121,...

which will divide n.