definition of principal 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 ideal of ring.
Target Context
- The reader will have a definition of principal 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:
\( R\): \(\in \{\text{ the rings }\}\)
\( p\): \(\in R\)
\(*I_l (p) \): \(= R p\), \(\in \{\text{ the left ideals of } R\}\)
\(*I_r (p)\): \(= p R\), \(\in \{\text{ the right ideals of } R\}\)
\(*I_b (p)\): \(= \{\sum_{j \in \{1, ..., k\}} (p_{l, j} p p_{r, j}) \vert k \in \mathbb{N}, p_{l, j} \in R \land p_{r, j} \in R\}\), \(\in \{\text{ the both-sided ideals of } R\}\)
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Conditions:
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\(I_l (p)\) is called "left principal ideal by \(p\)"; \(I_r (p)\) is called "right principal ideal by \(p\)"; \(I_b (p)\) is called "both-sided principal ideal by \(p\)" or "principal ideal by \(p\)".
2: Natural Language Description
For any ring, \(R\), and any element, \(p \in R\), \(I_l (p) := R p\) is called "left principal ideal by \(p\)"; \(I_r (p) := p R\) is called "right principal ideal by \(p\)"; \(I_b (p) := \{\sum_{j \in \{1, ..., k\}} (p_{l, j} p p_{r, j}) \vert k \in \mathbb{N}, p_{l, j} \in R \land p_{r, j} \in R\}\) is called "both-sided principal ideal by \(p\)" or "principal ideal by \(p\)"
3: Note
\(I_l (p)\) is indeed a left ideal: \(p_1 p + p_2 p = (p_1 + p_2) p \in I_l (p)\); \(0 = 0 p \in I_l (p)\); \(- (p_1 p) = (- p_1) p \in I_l (p)\); \(R R p = R p\).
\(I_r (p)\) is indeed a right ideal: \(p p_1 + p p_2 = p (p_1 + p_2) \in I_r (p)\); \(0 = p 0 \in I_r (p)\); \(- (p p_1) = p (- p_1) \in I_r (p)\); \(p R R = p R\).
\(I_b (p)\) is indeed a both-sided ideal: \(\sum_{j \in \{1, ..., k\}} (p_{l, j} p p_{r, j}) + \sum_{j \in \{1, ..., k'\}} (p'_{l, j} p p'_{r, j}) \in I_b (p)\); \(0 = 0 p 0 \in I_b (p)\); \(- (\sum_{j \in \{1, ..., k\}} (p_{l, j} p p_{r, j})) = \sum_{j \in \{1, ..., k\}} ((- p_{l, j}) p p_{r, j}) \in I_r (p)\); \(R I_b (p) = I_b (p) R = I_b (p)\).
We cannot define \(I_b (p)\) as \(R p R\), because \(p_{l, 1} p p_{r, 1} + p'_{l, 1} p p'_{r, 1}\) is not necessarily in \(R p R\).
When \(R\) is commutative, each left principal ideal or right principal ideal is a both-sided principal ideal, because for example, while \(I_l (p) \subseteq I_b (p)\) is obvious, for each \(\sum_{j \in \{1, ..., k\}} (p_{l, j} p p_{r, j})\), \(= \sum_{j \in \{1, ..., k\}} (p_{l, j} p_{r, j} p) = \sum_{j \in \{1, ..., k\}} (p_{l, j} p_{r, j}) p \in R p\), so, \(I_b (p) \subseteq I_l (p)\).
