A) Give an example to show that a factor ring of an integral domain may be a field
B) Give an example to show that a factor ring of an integral domain have divisors of 0.
C) Give an example to show that a factor ring of a ring with divisors of 0 may be integral domain.
A) $\mathbb{Q}[X]/\langle x+1\rangle$ and $\mathbb{Z}/4\mathbb{Z}$
B) $\mathbb{Z}/4\mathbb{Z}$,
C) $\mathbb{Z}_2/\mathbb{Z}_6$
Are my answers for A), B), C) correct? And can you give a few more examples for A), B) and C)?
There are general theorems about factor rings being fields and integral domains. Let $R$ be a commutative ring (with unity) and $I$ be an ideal.
Then there are two theorems:
1) $R/I$ is a field if and only if $I$ is maximal (i.e. there is no ideal $J$ satisfying $I\subset J\subset R$).
2) $R/I$ is an integral domain if and only if $I$ is prime (i.e. whenever $a,b\in R$ are such that $ab\in I$ then $a\in I$ or $b\in I$.)
You can now answer your questions generally:
A) To make $R/I$ a field you should take any ideal $I$ of $R$ that is maximal (by theorem $1$). For example $R=\mathbb{Z}$ and $I = p\mathbb{Z}$ give the fields $\mathbb{F}_p$ of integers mod $p$.
B) To make $R/I$ a non-integral domain you should take any ideal $I$ of $R$ that is NOT prime (by theorem $2$). For example $R=\mathbb{Z}$ and $I = m\mathbb{Z}$ where $m=ab$ is composite. The corresponding factor ring is the integers mod $m$ which is NOT an integral domain for our choice of $m$.
C) By theorem $1$ you just need to take $I$ to be any prime ideal. Fortunately prime ideals ALWAYS exist in any commutative ring (with unity) so really every such ring that is not an integral domain has an integral domain quotient! So finding examples should be easy enough.
Easy example, take $R = \mathbb{Z}/4\mathbb{Z}$. This is a ring but is NOT an integral domain, so fits the description of what we are looking for. Consider the prime ideal $I = \{0+4\mathbb{Z},2+4\mathbb{Z}\}$. Then $R/I = \{[0+4\mathbb{Z}],[1+4\mathbb{Z}]\}$ is an integral domain.
A) Take any irreducible polynomial $p(x)\in \mathbb{Q}[x]$, for example let $p(x)=x^2+1$, then $\mathbb{Q}[x]/p(x)$ is a field.
B) Take any reducible polynomial $p(x)\in \mathbb{Q}[x]$, for example let $p(x)=x^2-1$, then $\mathbb{Q}[x]/p(x)$ has a zero divisor.
C) $\mathbb{Z}_4$ has a zero divisor, but $\mathbb{Z}_4/2{\mathbb{Z}_4}$ is an integral domain.
– Zev Chonoles Aug 13 '13 at 12:28<and>mean "less than" and "greater than", and produce spacing correct for that meaning only; to make angle brackets, use\langleand\rangle.