# For which $n$, the $\mathbb Z_{n}$ for ring or Field. [closed]

Let $$\mathbb Z_{n}=\{0,1,...,n-1\}$$ and $$+_{n}$$ and $$\times_{n}$$ be the modulo addition and multiplication. For $$n=3$$, the set $$\mathbb Z_{n}$$ is not group wrt $$\times_{3}$$. I have the following questions.

(1) What should be the value of n (e.g., $$n\geq ?$$ ) so that $$Z_{n}$$ form ring with the above operations.

(2) I have saw that $$Z_{p}$$ form field for $$p$$ being a prime number. Now $$p=3$$ is prime but $$Z_{3}$$ doesn't form field.

• Why do you believe that $\Bbb Z_3$ is not a field? Mar 23 at 12:53
• Oh Thanks dear. I have forgot that the F-{0} should be abelian Mar 23 at 12:55
• Better consider it as the quotient ring $\Bbb Z/n\Bbb Z$ by the ideal $(n)=n\Bbb Z$. If $n$ is not prime then we have zero divisors. For $n=p$ prime this is not the case and we have a field. Mar 23 at 14:56

A commutative ring $$R$$ forms a field if the set $$R \setminus \{0\}$$ is a group under multiplication. The field $$\Bbb Z_3$$ satisfies this property: $$\Bbb Z_3 \setminus \{0\} = \{\pm 1\}$$ forms a group under multiplication.