07-04. Properties of Ideals

Definition.

• $R$,
  • ring
• $A \subseteq R$,

(1) The ideal generated by $A$, denoted by $(A)$, is the smallest ideal of $R$ containing $A$.

(2)

\[RA := \{ \sum_{i = 1}^{n} r_{i}a_{i} \mid r_{i} \in R, \ a_{i} \in A, \ n \in \mathbb{N} \} .\]

If $A = \emptyset$, $RA := 0$.

(3) An ideal generated by a single element is called a principal ideal. (4) An ideal generated by a finite set is called a finitedly generated ideal.

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Prooposition 9

• $R$,
  • ring
• $I \subseteq R$
  • ideal
• (1) $I = R$ iff $1 \in I$,
• (2) Assume $R$ is commutative. Then followinsgs are equivalent
  • (2-1) $R$ is a field
  • (2-2) its only ideals are $0$ and $R$.

proof

(1)

Suppose that $I = R$. Then $1 \in I = R$. Suppose that $1 \in I$.

\[\forall r \in R, \ r = r \cdot 1 \in I\]

$\Box$

Corollary 10.

proof

$\Box$

Definition maximal ideal

• $S$,
  • ring
• $M \subseteq S$,
  • ideal

$M$ is said to be maximal ideal if

• $M \neq S$,
• the only ideals containing $M$ are $M$ and $S$.

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Propostion 11.

• $R$,
  • ring with identity
• $J \subset R$,
  • maximal ideals

Then $J$ contains every proper ideals.

proof.

$\Box$

Proposition 12.

• $R$,
  • coomutative ring
• $M \subseteq R$,
  • ideal

Then the following statements are equivalent;

• (i) $M$ is maximal ideals
• (ii) $R/M$ is a field

proof.

By Proposition 9 (2) in Section 7.4, $R/M$ is a field if and only if ideals of $R/M$ are 0 and $R$. $M$ is maximal if and only if

$\Box$

Definition. prime ideal

• $R$,
  • commutative ring
• $P \subseteq R$,
  • ideal

$P$ is said to be prime ideal if $P \neq R$ and

\[\forall a, b \in R, \ ab \in P \Rightarrow ( a \in R \text{ or } b \in R )\]

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Proposition 13.

• $R$,
  • commutative ring
• $P$,
  • ideal

Then the following statements are equivalent;

• (1) $P$ is a prime ideal in $R$
• (2) The quotient ring $R/P$ is an integral domain

proof.

Let $[r]_{P} \in R/P$. $r \in P$ if and only if \([r]_{P} = [0]_{P}\). From above obvervation, The following statements are equivalent respectively.

• $P \neq R$
• \([r]_{P} \neq [0]_{P}\),

and

• If $ab\in P$, $a \in P$ or $b \in P$.
• If \([ab]_{P} = [0]_{P}\), \([a]_{P} = 0\) or \([b]_{P} = [0]\).

$\Box$

Reference