Equivalence of Definitions of Commutative Local Ring

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $A$ be a commutative ring with unity.

The following definitions of the concept of Commutative Local Ring are equivalent:

Definition 1

The ring $A$ is local if and only if it has a unique maximal ideal.

Definition 2

The ring $A$ is local if and only if it is nontrivial and the sum of any two non-units is a non-unit.

Definition 3

Let $M \subseteq A$ be the subset of the non-units of $A$.

The ring $A$ is local if and only if $M$ is a proper ideal of $A$.


Proof

Definition 1 implies Definition 2

Let $\mathfrak m \subsetneq A$ be the unique maximal ideal.

First, $A$ is nontrivial, since $1 \notin \mathfrak m$.

Secondly, let $x, y \in A$ be non-units.

Let $\ideal x$ and $\ideal y$ be the principal ideals generated by $x$ and $y$, respectively.

In view of the unique maximality, $\ideal x \subseteq \mathfrak m$ and $\ideal y \subseteq \mathfrak m$.

In particular:

$x, y \in \mathfrak m$

which implies:

$x + y \in \mathfrak m$

Therefore $x + y$ is a non-unit.

$\Box$


Definition 2 implies Definition 3

First, by the assumption:

$x, y \in M \implies x + y \in M$

Secondly, let $a \in A$ and $x \in M$.

We shall show that $a x \in M$.

Aiming for a contradiction, suppose there exists a $u \in A$ such that:

$u \paren {a x} = 1$

Thus:

$\paren {u a} x = 1$

which means that $x$ is a unit.

This contradicts the fact that $x \in M$.

Therefore $a x \in M$.

Altogether $M$ is an ideal.

Finally, $1 \notin M$, since $A$ is nontrivial.


This article, or a section of it, needs explaining.
In particular: Why?
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.


Therefore $M$ is a proper ideal.

$\Box$


Definition 3 implies Definition 1

Let $I$ be an arbitrary ideal such that $M \subsetneq I$.

Let $u \in I \setminus M$.

Then $u$ is a unit by the definition of $M$.

Therefore $M = A$.

Furthermore, $M$ is assumed to be a proper ideal.

Therefore $M$ is a maximal ideal.

On the other hand, let $N$ be an arbitrary maximal ideal.

As $N$ does not contain a unit:

$N \subseteq M$

which implies:

$N = M$

Therefore $M$ is the unique maximal ideal.

$\blacksquare$

Retrieved from "https://proofwiki.org/w/index.php?title=Equivalence_of_Definitions_of_Commutative_Local_Ring&oldid=584666"