0

Let $R$ is a commutative ring which has no proper ideals. Prove that $R$ is a field.

Linkin
  • 35
  • 5
    There is already an answer to this question!! Here is the thread http://math.stackexchange.com/questions/101157/a-ring-is-a-field-iff-the-only-ideals-are-0-and-1 – Josh R May 22 '16 at 09:14
  • 1
    Counterexample: the trivial ring. – sTertooy May 22 '16 at 09:20

1 Answers1

2

Take $x\in R$ and consider the ideal $(x)$ generated by $x$. Then $(x)=R$ and hence $1\in (x)$, it follows that there exists a $y\in R$ such that $xy=1=yx$. Thus $y=x^{-1}$. Since $x$ was chosen arbitrarily, the result follows.

Mathematician 42
  • 12,591