- #1

cbarker1

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MHB

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I am reading the Abstract Algebra Book by Dummit and Foote. I am confusing with this example for one-side ideals. So here is the example:

Let $R$ be a commutative ring with $1 \ne 0$ and let \( n\in \mathbb{Z} \) with $n\ge 2$. For each $j\in \{1,2,\dots, n\}$, let $L_j$ be the set of all $n \times n$ matrices in $M_n(R)$ with arbitrary entries in the jth column and zeroes in all other columns. It is clear that $L_j$ is closed under subtraction. It follows directly from the definition of matrix multiplication that any matrix $T \in M_n(R)$ and $A \in L_j$, the product $TA$ has zero entries in the ith column for all $i\ne j$. This shows $L_j$ is a left ideal of $M_n(R)$. Moreover, $L_j$ is not a right ideal.

What does this example look like with math symbols?

Thanks,

Cbarker1