202505211605

Tags : Ring Theory

Prime Ideals and Maximal Ideals

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Definition

Let $I$ be a proper ideal of the commutative ring $R$.

• $I$ is called a Prime Ideal if $R/I$ is an Integral Domain
• $I$ is called a Maximal Ideal if $R/I$ is a Field

The names do make sense because they satisfy the following property:

Lemma

Given a proper ideal $I$ of a commutative ring $R$

• $I$ is prime iff for all $a,b\in R$ we have $ab=I$ iff $a\in I$ or $b\in I$
• $I$ is maximal iff for all ideals $J$ such that $I\subseteq J$ we have $I=J$ or $J=R$.

By Finite Integral Domains are Fields, we get the following.

Lemma

If $I$ a prime ideal of $R$ such that $R/I$ is finite, then $I$ is a Maximal Ideal.

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References

• Integral Domain
• Fields
• Finite Integral Domains are Fields