User:Fake Proof/Sandbox/Generated Ideal of Ring

Definition
Let $\struct {R, +, \circ}$ be a ring.

Let $\O \subsetneq S \subseteq R$.

Definition 1
The ideal generated by $S$ is the smallest ideal containing $S$.

Definition 2
The ideal generated by $S$ is the intersection of all ideals containing $S$.

Definition 3
Let $\gen S$ be the additive subgroup of $\struct {R, +}$ generated by $S$.

Let $R \circ S$ be the set of left-sided linear combinations of $S$. That is:


 * $\ds R \circ S = \set {\sum_{i \mathop = 1}^n r_i \circ s_i: n \in \N, r_i \in R, s_i \in S}$

Let $S \circ R$ be the set of right-sided linear combinations of $S$. That is:


 * $\ds S \circ R = \set {\sum_{i \mathop = 1}^n s_i \circ r_i: n \in \N, r_i \in R, s_i \in S}$

Let $R \circ S \circ R$ be the set of two-sided linear combinations of $S$. That is:


 * $\ds R \circ S \circ R = \set {\sum_{i \mathop = 1}^n a_i \circ s_i \circ b_i: n \in \N, a_i, b_i \in R, s_i \in S}$

The ideal generated by $S$ is $\gen S + R \circ S + S \circ R + R \circ S \circ R$.

Definition 1
The left ideal generated by $S$ is the smallest left ideal containing $S$.

Definition 2
The left ideal generated by $S$ is the intersection of all left ideals containing $S$.

Definition 3
Let $\gen S$ be the additive subgroup of $\struct {R, +}$ generated by $S$.

Let $R \circ S$ be the set of left-sided linear combinations of $S$. That is:


 * $\ds R \circ S = \set {\sum_{i \mathop = 1}^n r_i \circ s_i: n \in \N, r_i \in R, s_i \in S}$

The left ideal generated by $S$ is $\gen S + R \circ S$.

Definition 1
The right ideal generated by $S$ is the smallest right ideal containing $S$.

Definition 2
The right ideal generated by $S$ is the intersection of all right ideals containing $S$.

Definition 3
Let $\gen S$ be the additive subgroup of $\struct {R, +}$ generated by $S$.

Let $S \circ R$ be the set of right-sided linear combinations of $S$. That is:


 * $\ds S \circ R = \set {\sum_{i \mathop = 1}^n s_i \circ r_i: n \in \N, r_i \in R, s_i \in S}$

The right ideal generated by $S$ is $\gen S + S \circ R$.

Also defined as
Some sources define the ideal generated by $S$ as $R \circ S \circ R$. It is also an ideal of $\struct {R, +, \circ}$, but different in general.

Also see

 * Equivalence of Definitions of Generated Ideal of Ring
 * Definition:Generator of Ideal of Ring
 * Definition:Generated Subring
 * Generated Ideal of Ring is Closure Operator