definition of ideal of ring
Topics
About: ring
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of ring.
Target Context
- The reader will have a definition of ideal of ring.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( R\): \(\in \{\text{ the rings }\}\)
\(*I_l\): \(\in \{\text{ the additive subgroups of } R\}\)
\(*I_r\): \(\in \{\text{ the additive subgroups of } R\}\)
\(*I_b\): \(\in \{\text{ the additive subgroups of } R\}\)
//
Conditions:
\(R I_l = I_l\)
\(\land\)
\(I_r R = I_r\)
\(\land\)
\(R I_b = I_b R = I_b\)
//
\(I_l\) is called "left ideal"; \(I_r\) is called "right ideal"; \(I_b\) is called "both-sided ideal" or "ideal".
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"
3: Note
The condition, \(R I_l = I_l\), is equivalent with the condition, \(\forall p_1 \in R, \forall p_2 \in I_l (p_1 p_2 \in I_l)\), 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 \(\forall p_1 \in R, \forall p_2 \in I_r (p_2 p_1 \in I_r)\).
Likewise, \(R I_b = I_b R = I_b\) is equivalent with \(\forall p_1 \in R, \forall p_2 \in I_b (p_1 p_2 \in I_b \land p_2 p_1 \in I_b)\).
When \(R\) is commutative, each left ideal or right ideal is a both-sided ideal.
