Give an example to show that a factor ring of an integral domain may be a field

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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.

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Updated on August 01, 2022

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  • user88310
    user88310 over 1 year

    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)?

    • Daniel Fischer
      Daniel Fischer about 10 years
      In A), you haven't said what ideal to factor out, and the $\mathbb{Z}/(4\mathbb{Z})$ surely is a typo, as that's your example for a factor ring with zero divisors. B) is fine, C) doesn't make sense, I suppose you meant $\mathbb{Z}_6/(2\mathbb{Z}_6)$.
    • walcher
      walcher about 10 years
      Try to prove: $R/I$ is a field $\Leftrightarrow I$ is a maximal ideal. $R/I$ is an integral domain $\Leftrightarrow I$ is a prime ideal (that means if $ab \in I$, then $a \in I$ or $b \in I$). That should clear it up.
    • Zev Chonoles
      Zev Chonoles about 10 years
      @user88310: Please see here for a guide to writing math with MathJax, and see here for a guide to formatting posts with Markdown. Some MathJax advice: < and > mean "less than" and "greater than", and produce spacing correct for that meaning only; to make angle brackets, use \langle and \rangle.