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:

\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:

\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. |