Examples of Modules - Left Ideals J are Left X-Modules

Examples of Modules - Left Ideals J are Left X-Modules

Let $X$ be an algebra and let $J$ be a left-ideal of $X$. Then $XJ \subseteq J$. Let $f : X \times J \to J$ be the module multiplication defined by $f(x, j) = xj$, that is, the product is simply the multiplication given by the multiplication in $X$.

For each fixed $x \in X$ the mapping $f_x : J \to J$ defined by $f_x(j) = xj$ is clearly linear, since if $j_1, j_2 \in J$ then $x(j_1 + j_2) = xj_1 + xj_2$ by distributivity in $X$. So axiom $LM1$ is satisfied.

For each fixed $j \in J$ the mapping $f_j(x) : X \to J$ defined by $f_j(x) = xj$ is also linear since if $x_1, x_2 \in X$ then $(x_1 + x_2)j = x_1j + x_2j$, again by distributivity in $X$. So axiom $LM2$ is satisfied.

Lastly, for all $x_1, x_2 \in X$ and all $j \in J$ we certainly have by the associativity of multiplication in $X$ that:

(1)
\begin{align} \quad x_1(x_2j) = (x_1x_2)j \end{align}

So axiom $LM3$ is satisfied. Thus any left-ideal $J$ of $X$ is a left $X$-module. In a similar fashion it can be shown that any right ideal of $X$ is a right $X$-module, and that any two-sided ideal of $X$ is an $X$-bimodule.

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