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I have a question regarding the calculation of right and left ideals of a matrix ring. I understand the concept, or so I believe, but, when consulting the argument on this page http://mathworld.wolfram.com/RightIdeal.html I harbour some doubts I would like to share with you. The calculation of the matrix product proving the right ideal is crystal-clear to me. However, when dealing with the argument for excluding the possibility of that very same structure being a left ideal, there is something I do not fully understand.

1) Does it mean the matrix belonging to the Ideal, when being multiplied by another matrix, has to occupy the inverse position to the kind of ideal we are dealing with (so that if we are talking about a right ideal, it will be the left-sided member of the noncommutative matrix product, and when talking about an alleged left ideal it hast to be on the right of the product)?

2) If 1) is the way I suggest, and given the counterexample in (6) on the link, is it the case, as I suspect, that by inverting the order of the product members in (6) we once again get an ideal, namely a matrix with all entries zero?

Could you please confirm me these points?

Thanks in advance.

Javier Arias
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1 Answers1

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The point is that for $A\in\{\left[\begin{smallmatrix}a&b\\0&0\end{smallmatrix}\right]\mid a, b\in \Bbb R\}=T$, and any other matrix $M$ in $R=M_2(\Bbb R)$, you have $AM\in T$, but the example in (6) demonstrates an example of an $A\in T$ and an $M$ such that $MA\notin T$.

So as sets $TR\subseteq T$, but $RT\nsubseteq T$. In words, $T$ absorbs multiplication on the right by elements of $R$, but not on the left.

..that by inverting the order of the product members in (6) we once again get an ideal, namely a matrix with all entries zero?

You will not get an ideal but you will get a matrix, and yes, this matrix will be inside the right ideal.

rschwieb
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  • Thanks for your reply to old posts of mine. I appreciate it very much that you took the time and effort, given that sometimes I have the feeling I am just uttering nonsense...... It is difficult for me to articulate the questions at this level..... – Javier Arias Sep 11 '15 at 19:08
  • @JavierArias I understand. As long as you are responsive to feedback, I'm sure you'll enjoy better questions, answers and positive community attention. – rschwieb Sep 12 '15 at 00:31