Exploring Right Ideals in M(n,n)

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In summary: D? or B?... i forget...) that W is the set of all these matrices.In summary, the conversation discusses a problem involving a subspace R of a vector space V=M(n,n) and a subspace W of R_n. The question involves showing certain properties and uses the concept of "right ideals". The conversation also includes a discussion of the notation used in the problem and offers hints and explanations for better understanding it.
  • #1
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"right ideals"

Ok, so this is an extra credit question on a test, i haven't really tried it yet, but the test is thurs, so i figured i'd try to post this to see what anyone says, and then see what i work out, or whatever. I don't even know what "right ideals" means, but our prof said that's what the question was about... so i figured... ya...

Let R be a subspace of V = M(n,n) such that AB is in R whenever A is in R. Let W be the subspace of R_n spanned by all AX with A in R, X in R_n.

A) show that for any matrix A in M
i) Aej = Aj and ii) A= summation(AjeJ)

B) show that if AX is in W for every X in R_n then Aj is in W.

C) Write Aj as a linear combination of products A(ij)X(ij), A(ij) in R, X(ij) in R_n

D) use ii) to show that if A is as in B), then A is in R
E) show that R consists of all matrices A in M with AX in W for all X in R_n

OOOOK... so that's the problem. The only hint he gave us was the "right ideals" thing. So i'll google that and see if i can work this all out in the morning. I pretty much have tons of trouble with this stuff though, so we'll see. Any help would be totally awesome! oh, and if you don't understand the question... join the club... i can try to explain his notation if its weird... but that's about it. thanks in advance...
 
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  • #2
I'm boggled by most of the notation -- you probably should explain what all of it means. (The only thing I'm sure of is that M(n, n), or [itex]M_{n, n}[/itex] means the vector space of nxn matrices, presumably with real entries!)


I can make a comment that I think will be useful -- if you write the identity matrix, I as a sum of other matrices, then it is sometimes productive to observe that A = AI, and then replace I with that sum... and then later use this "decomposition" by substuting it in for A.
 
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  • #3
mk, i'll try and rewrite in latex, even though our test is written entirely this way, i'll do my best to explain...

Let R be a subspace of [tex]V = M_{n,n}[/tex] such that AB is in R whenever A is in R. Let W be the subspace of [tex] R_{n} [/tex] spanned by all AX with A in R, X in [tex] R_{n}. [/tex]

a)show that for any matrix A in M
i) Aej = Aj and ii) [tex] A= \Sigma (Ajej) [/tex]

[ok, so here, ej means the the j-ith column of the identity matrix... i think... see if that makes sense. Aj is the jth column of A i guess, and the second ej could be the jth row of the identity or jth column... I'm not sure]

b) ...
[same as before, i guess R_n means [tex]R_{n} [/tex] if that helps... Aj means jth column of j i assume...]

c) again.. tex doesn't change anything...
[A(ij) is the entry of A in the ith row and jth column. X(ij) is the entry in X in the ith row jth column.]

d) and e) stay the same, i don't know how i could explain those any better...

Anyways, ya, he's notation is sort of weird i guess... but i don't have a book to really notice the difference anyways. So, maybe that helps... again, i'll do out some of the work if i can in the morning. Its way to late to make sense of this right now.
 
  • #4
I don't know what R_n means, though.

C) is odd -- Aj is a matrix, but if A_ij and X_ij are scalars... no linear combination of scalars can be a matrix. My best guess is he's again doing something weird... instead of A_ij being the (i,j)-th entry of A, he's saying that A_ij is just yet another matrix. (because he says A(ij) in R) I guess that's why he wrote A(ij) instead of A_ij?
 
  • #5
R_n just means numbers in the nth dimension.
 
  • #6
I'm not so sure that X(ij) is a scaler. I think that X(ij) is just the vector that goes with the scaler A(ij). The question does say that X(ij) is in R_n which suggests we are talking about a vector.

A right ideal (in the case of vectors) is exactly the definition of R. ie a vector subspace of a ring V(where you consider the ring a vector space over its centre) with the property that if [tex]A\in R[/tex] and [tex]B\in R[/tex] then [tex]AB \in R[/tex].

The only thing to watch out for from parts A through D is that you show some care to recognize when R means the right ideal and when R means the reals. Part A section ii is a little bit of a mystery to me, even if you take ej to be a row vector.

B is simple. You just need to consider the right X in R_n. Hint you use these vectors all the time. You've even used them in part A

Again in C you just need to choose the right vector for X(ij)

D is a little trickier. I might be finding it tricky because I don't understand what ii) means. First can you show Aj is in W? Then consider what it means to be in W (ie the span of some set). With that in mind can you right out Aj as a sum of elements of the form Bx with B in right ideal and x some vector. Now can you right out A as a sum of Matrices of the form BX with B in the right ideal and X just some matrix. What does that mean about A?

And then E becomes easy again. If A is in R does it satisfy the condition? And you showed in D that if A satisfies the condition then A is in R.

I don't think I've given too much away and I hope I've cleared up the problem a little.

Steven
 

1. What are right ideals in M(n,n)?

Right ideals in M(n,n) are subsets of the matrix algebra M(n,n) that satisfy the property that if a matrix A is in the right ideal, then the product of A with any other matrix B in M(n,n) is also in the right ideal.

2. How do we explore right ideals in M(n,n)?

One way to explore right ideals in M(n,n) is to analyze the structure of the matrices that belong to the right ideal. This can provide insight into the properties and characteristics of the right ideal.

3. What is the significance of exploring right ideals in M(n,n)?

Exploring right ideals in M(n,n) can help us understand the algebraic structure of matrices and their relationships. It can also have implications in various fields such as linear algebra, representation theory, and algebraic geometry.

4. How do right ideals relate to other mathematical concepts?

Right ideals in M(n,n) have connections to other mathematical concepts such as ideals in ring theory, subspaces in linear algebra, and modules in abstract algebra. Understanding these connections can provide a deeper understanding of the properties of right ideals.

5. Can right ideals be explored in other matrix algebras?

Yes, the concept of right ideals can be extended to other matrix algebras, such as M(m,n) where m and n are not necessarily equal. However, the properties and structures of right ideals may differ in these algebras compared to M(n,n).

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