Question

Prove that Z32 with the operations of [+] and [*] as defined below is not an integral domain.

The set of integers mod m is denoted Zm. The elements of Zm are denoted [x]m where x is an integer from 0 to m-1. Each element [x]m is an equivalence class of integers that have the same integer remainder as x when divided by m.

For example, Z7 = {[0]7, [1]7, [2]7, [3]7, [4]7, [5]7, [6]7}. The element [5]7 represents the infinite set of integers of the form 5 plus an integer multiple of 7. That is [5]7 = {. . . –9, –2, 5, 12, 19, 26, . . .} or, more formally, [5]7 = {y: y = 5 + 7q for some integer q}.

Modular addition, [+], is defined on the set Zm in terms of integer addition as follows: [a]m [+] [b]m equals the element of Zm that contains the integer a+b.

Modular multiplication, [*], is defined on the set Zm in terms of integer multiplication as follows: [a]m [*] [b]m equals the element of Zm that contains the integer ab.

The set Zm with these operations is a ring for any integer m.

Answer 1

By the definition of Integral Domain.

Z₃₂ is an integral domain if its a ring and for any ab = 0 implies either a = 0 or b = 0.

As it is a ring defined on operation + and * we shall now show it is not an Integral Domain.

We know

Thus, Z₃₂ has zero divisors and hence it is not an Integral Domain.