Existence of a Maximal Ideal in an Artinian Ring

Question

How do we show that every Artinian Ring has at least one Maximal Ideal?

Answer 1

Any ring $R$ with unit has a maximal ideal. Indeed, consider the set $\mathcal{P}$ of proper ideals of $R$ and partially order this via the inclusion $\subseteq $. Since $\{0\}$ is in this poset, we get a non-empty poset.

Take any chain of proper ideals $\{I_\alpha\}_{\alpha \in A}$in the poset $\mathcal{P}$. The union of these ideals is again a proper ideal, as I demonstrated in this post:

Prove that $I=\bigcup_{\alpha\in A} I_{\alpha}$ is a proper ideal of $R$

Hence, any chain in $\mathcal{P}$ has an upper bound. Zorn's lemma yields a maximal element, and this is the maximal ideal you are looking for.

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Perhaps, you meant to ask about minimal ideal. Recall that an ideal is called minimal if it is non-zero and does not properly contain a non-zero ideal.

It is true that Artinian rings posses such ideals. Here is why.

Consider the ideal $R$. If this does not contain a proper non-zero ideal, then $R$ is minimal and we are done. Otherwise, there is an ideal $I_1$ such that $0 \neq I_1 \subsetneq R$. If $I_1$ does not contain a proper non-zero ideal, then $I_1$ is minimal. Otherwise, there is an ideal $I_2$ such that $0 \neq I_2 \subsetneq I_1$. Continue this process, if necessary. Eventually, it will have to stop, giving us a minimal ideal. Otherwise we get a chain

$$I_1 \supsetneq I_2 \supsetneq I_3 \supsetneq \dots$$

which contradicts that $R$ is Artinian.

Thus any Artinian ring has a minimal ideal.

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It is not true that arbitrary rings have minimal ideals. For example the ring $\mathbb{Z}$ has no minimal ideals.