Ideals in a Linear Space

Ideals in a Linear Space

Ideals

Definition: Let $X$ be a linear space and let $J \subseteq X$ be a linear subspace of $X$.
1) $J$ is said to be a Left Ideal of $X$ if $XJ \subseteq J$.
2) $J$ is said to be a Right Ideal of $X$ if $JX \subseteq J$.
3) $J$ is said to be a Two-Sided Ideal (or Bi-Ideal) of $X$ if it is both a left ideal of $X$ and a right ideal of $X$.

Here, $XJ$ denotes the set $XJ = \{ xj : x \in X, j \in J \}$ and $JX$ denotes the set $JX = \{ jx : j \in J, x \in X \}$.

Proposition 1: Let $X$ be an algebra and let $J \subseteq X$. If $J$ is a left/right/two-sided ideal of $X$ then $J$ is a subalgebra of $X$.
  • Proof: We only prove the case when $J$ is a left ideal. The cases when $J$ is a right ideal and a two-sided ideal are analogous.
  • Since $J$ is a left ideal of $X$ we have that $J$ is a linear subspace of $X$ and so it closed under the operations of addition and scalar multiplication and contains the zero vector. Also, since $J$ is a left ideal of $X$ we have that $XJ = \{ xj : x \in X, j \in J \} \subseteq J$, and so:
(1)
\begin{align} \quad JJ = \{ j_1j_1 : j_1, j_1 \in J \} \subseteq J \end{align}
  • Which shows that $J$ is closed under the operation of multiplication. Thus $J$ is a subalgebra of $X$. $\blacksquare$

Proper Ideals and Maximal Ideals

Definition: Let $X$ be a linear space and let $J \subseteq X$ be a (left/right/two-sided) ideal of $X$. Then $J$ is said to be a Proper (Left/Right/Two-Sided) Ideal of $X$ if $J \neq X$.
Definition: Let $X$ be a linear space and let $J \subseteq X$ be a (left/right/two-sided) ideal of $X$. Then $J$ is said to be a Maximal (Left/Right/Two-Sided) Ideal of $X$ if $J$ is a proper (left/right/two-sided) ideal of $X$ and if $K$ is any other proper (left/right/two-sided) ideal of $X$ then $J \not \subseteq K$.

Modular Ideals and Maximal Modular Ideals

Definition: Let $X$ be a linear space and let $J \subseteq X$ be a linear subspace of $X$.
1) A point $u \in X$ is said to be a Left Modular Unit of $J$ if $(1 - u)X \subseteq J$.
2) A point $u \in X$ is said to be a Right Modular Unit of $J$ if $X(1 - u) \subseteq J$.
3) A point $u \in X$ is said to be a Two-Sided Modular Unit of $J$ if it is both a left modular unit of $J$ and a right modular unit of $J$.

Here, $(1 - u)J$ denotes the set $(1 - u)X = \{ x - ux : x \in X \}$ and $X(1 - u)$ denotes the set $X(1 - u) = \{ x - xu : x \in X \}$.

Proposition 2: Let $X$ be an algebra and let $J \subseteq X$ be a linear subspace of $X$. If $X$ is an algebra with unit then $1$ is a left/right/two-sided modular unit of $J$.
  • Proof: If $X$ is an algebra with unit $1$ then since $J$ is a subspace (and hence clearly contains the zero vector) we have that:
(2)
\begin{align} \quad (1 - 1)X = \{ x - x : x \in X \} = \{ 0 \} \subseteq J \end{align}
(3)
\begin{align} \quad X(1 - 1) = \{ x - x : x \in X \} = \{ 0 \} \subseteq J \end{align}
  • So $1$ is a left/right/two-sided modular unit of $J$. $\blacksquare$
Definition: Let $X$ be a linear space and let $J \subseteq X$ be a linear subspace of $X$.
1) $J$ is said to be a Left Modular Ideal if $J$ is a left ideal and has a right modular unit.
2) $J$ is said to be a Right Modular Ideal if $J$ is a right ideal and has a left modular unit.
Definition: Let $X$ be a linear space and let $J \subseteq X$ be a (left/right/two-sided) ideal of $X$. Then $J$ is said to be a Maximal Modular (Left/Right/Two-Sided) Ideal of $X$ if $J$ is a proper (left/right/two-sided) ideal of $X$; if $K$ is any other proper (left/right/two-sided) ideal of $X$ then $J \not \subseteq K$; and $J$ is a (left/right/two-sided) modular ideal.
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