90: Ideal of Ring

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definition of ideal of ring

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Topics

About: ring

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of ring.

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Target Context

• The reader will have a definition of ideal of ring.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$R$: $\in \{\text{ the rings }\}$
$\ast I_l$: $\in \{\text{ the additive subgroups of }R\}$
$\ast I_r$: $\in \{\text{ the additive subgroups of }R\}$
$\ast I_b$: $\in \{\text{ the additive subgroups of }R\}$
//

Conditions:
$R{I}_l=I_l$
$\wedge$
$I_{r}R=I_r$
$\wedge$
$R{I}_b=I_{b}R=I_b$
//

$I_l$ is called "left ideal"; $I_r$ is called "right ideal"; $I_b$ is called "both-sided ideal" or "ideal".

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2: Natural Language Description

For any ring, $R$, any additive subgroup, $I_l\subseteq R$, such that $R{I}_l=I_l$ is called "left ideal"; any additive subgroup, $I_r\subseteq R$, such that $I_{r}R=I_r$ is called "right ideal"; any additive subgroup, $I_b\subseteq R$, such that $R{I}_b=I_{b}R=I_b$ is called "both-sided ideal" or "ideal"

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3: Note

The condition, $R{I}_l=I_l$, is equivalent with the condition, $\mathrm{\forall }{p}_1\in R,\mathrm{\forall }{p}_2\in I_l(p_1{p}_2\in I_l)$, because supposing the former, the latter is obviously satisfied; supposing the latter, $R{I}_l\subseteq I_l$, but as $1\in R$, $I_l\subseteq R{I}_l$, so, $R{I}_l=I_l$: we have adopted the definition of ring that requires the existence of the multiplicative identity.

Likewise, $I_{r}R=I_r$ is equivalent with $\mathrm{\forall }{p}_1\in R,\mathrm{\forall }{p}_2\in I_r(p_2{p}_1\in I_r)$.

Likewise, $R{I}_b=I_{b}R=I_b$ is equivalent with $\mathrm{\forall }{p}_1\in R,\mathrm{\forall }{p}_2\in I_b(p_1{p}_2\in I_b\wedge p_2{p}_1\in I_b)$.

When $R$ is commutative, each left ideal or right ideal is a both-sided ideal.

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References

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