Understanding One-Sided Ideals in Mathematics

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In summary, The matrix $A$ in the left ideal $L_{j}$ would be written as $A = \begin{bmatrix} 0 & \ldots & a_{1j} & \ldots & 0\\ 0 & \ldots & a_{2j} & \ldots & 0\\ \vdots & \ddots & \vdots & \ddots & \vdots\\ 0 & \ldots & a_{nj} & \ldots & 0\end{bmatrix},$ and the matrix $T\in M_{n}(R)$ would have the form $T = \begin{bmatrix} t_{11} & \ldots
  • #1
cbarker1
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Dear Everyone,

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
 
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  • #2
Hi Cbarker1,

It's not entirely clear to me what your question is exactly. If you mean you'd like to see verification of the left ideal/non-right ideal claim, that would look something like this.

Proof $L_{j}$ is a Left Ideal
Let $A\in L_{j},$ $T\in M_{n}(R),$ and let $a_{j}$ be the $j$th column of $A$. Then $$TA = T\left[\begin{array}{c|c|c|c|c} 0 & \ldots & a_{j} & \ldots & 0 \end{array}\right] = \left[\begin{array}{c|c|c|c|c} 0 & \ldots & Ta_{j} & \ldots & 0 \end{array}\right],$$ which shows that $TA\in L_{j}$. Hence, $L_{j}$ is a left ideal over $M_{n}(R).$

Proof $L_{1}$ is not a Right Ideal
Let $a_{1} = \begin{bmatrix}1\\ 0\\ \vdots\\ 0 \end{bmatrix},$ $A = \left[\begin{array}{c|c|c|c} a_{1} & 0 &\ldots & 0 \end{array} \right],$ and $T = \left[\begin{array}{c|c|c|c|c} a_{1} & a_{1} & 0 &\ldots & 0 \end{array} \right].$ Then $AT = T\notin L_{1}.$ Hence, $L_{1}$ is not a right ideal over $M_{n}(R).$ This example can be generalized to any $j$, if desired.
 
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  • #2
I am trying to see the symbols. Like this: \[ \begin{pmatrix} 1 & 2 & 3\\ a & b & c \end{pmatrix} \].
 
  • #3
I am trying to see the symbols. Something Like this: \[ \begin{pmatrix} 1 & 2 & 3\\ a & b & c \end{pmatrix} \].
 
  • #4
OK, let's see if this helps.

A matrix in $L_{j}$ would look be written as $$A = \begin{bmatrix} 0 & \ldots & a_{1j} & \ldots & 0\\ 0 & \ldots & a_{2j} & \ldots & 0\\ \vdots & \ddots & \vdots & \ddots & \vdots\\ 0 & \ldots & a_{nj} & \ldots & 0\end{bmatrix},$$ and a matrix $T\in M_{n}(R)$ would have the form $$T = \begin{bmatrix} t_{11} & \ldots & t_{1j} & \ldots & t_{1n}\\ t_{21} & \ldots & t_{2j} & \ldots & t_{2n}\\ \vdots & \ddots & \vdots & \ddots & \vdots\\ t_{n1} & \ldots & t_{nj} & \ldots & t_{nn}\\\end{bmatrix},$$ where all the $a$'s and $t$'s are elements of $R$. Using these forms for $A$ and $T$, the product $TA$ would be $$TA = \begin{bmatrix}0 & \ldots & t_{11}a_{1j} + t_{12}a_{2j}+ \ldots + t_{1n}a_{nj} & \ldots & 0\\ 0 & \ldots & \vdots & \ldots & 0\\ \vdots & \ddots & \vdots & \ddots & \vdots\\ 0 & \ldots & \ldots & \ldots & 0 \end{bmatrix},$$ where I have left rows $2$ through $n$ of the $j$th column of $TA$ for you to try to fill in for yourself.

Does this answer your question?
 
  • #4
yes.
 

FAQ: Understanding One-Sided Ideals in Mathematics

1. What is a one-sided ideal in mathematics?

A one-sided ideal is a subset of a ring that is closed under multiplication on one side, either left or right. It is a generalization of an ideal, which is closed under multiplication on both sides.

2. How are one-sided ideals different from two-sided ideals?

One-sided ideals are only closed under multiplication on one side, while two-sided ideals are closed under multiplication on both sides. This means that a one-sided ideal may not contain all possible products of elements in the ring, whereas a two-sided ideal does.

3. What are some examples of one-sided ideals?

Examples of one-sided ideals include the set of all multiples of a fixed element in a ring, the set of all polynomials with a fixed constant term, and the set of all matrices with a fixed column or row.

4. How do one-sided ideals relate to quotient rings?

One-sided ideals are used to construct quotient rings, which are rings formed by taking the cosets of a given ideal. In a quotient ring, the elements are equivalence classes of elements in the original ring, where two elements are equivalent if their difference is in the ideal.

5. What is the significance of understanding one-sided ideals in mathematics?

Understanding one-sided ideals is important in abstract algebra, as they provide a way to study the structure of rings and quotient rings. They also have applications in algebraic geometry, functional analysis, and other areas of mathematics.

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