What is the Ring Generated by R-Linear Combinations of These Matrices?

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Discussion Overview

The discussion revolves around the classification and properties of a set of four matrices, specifically whether they form a ring and what that ring might be called. Participants explore concepts related to closure under addition and multiplication, as well as the implications of these properties in the context of algebraic structures.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants assert that the set of matrices is closed under multiplication, questioning its classification.
  • One participant suggests that the set might be the Klein 4-group, referencing external material for clarification.
  • Another participant emphasizes the need for the set to be a ring for closure, seeking to understand its name.
  • Concerns are raised about the absence of a zero element and the definitions of addition and multiplication operations in this context.
  • A participant claims that the set is not closed under addition, thus disqualifying it as a ring.
  • There is a suggestion to consider the ring generated by the R-linear combinations of the matrices, implying a broader context of matrix algebra.
  • One participant points out that the last matrix squared does not belong to the group, indicating a potential issue with closure under multiplication.
  • Another participant states that if the matrices are indeed considered over R, they generate all of M_2(R), the ring of 2x2 matrices.

Areas of Agreement / Disagreement

Participants express disagreement regarding the classification of the set as a ring, with some asserting it is not a ring due to lack of closure under addition, while others propose that it may generate a larger structure. The discussion remains unresolved with competing views on the properties of the matrices.

Contextual Notes

There are limitations regarding the assumptions about operations defined on the matrices, and the discussion does not resolve the mathematical steps necessary to establish closure under addition or multiplication.

plxmny
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This set of four matrices:

1 0 | 0 1 | -1 0 | 0 1
0 1 | 1 0 | 0 1 | -1 0

are closed under multiplication. What is it called? I know that it is not
those silly quaternions
 
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plxmny said:
This set of four matrices:

1 0 | 0 1 | -1 0 | 0 1
0 1 | 1 0 | 0 1 | -1 0

are closed under multiplication. What is it called? I know that it is not
those silly quaternions

Is it the Klein 4-group?
http://en.wikipedia.org/wiki/Klein_four-group
 
Thanks, NateTG, for taking the time to respond. That is what I thought until I realized that it had to be a RING for closure. So then I thought I would like to know the NAME of it, at least.
 
If it's a ring, where's zero, and what are the addition and multiplication operations?
 
NateTG,

0 is just

0 0
0 0

and + is just matrix addition and x is just matrix multiplication.

I found some stupid wikipedia thing where these 4 were given names like
K0 K1 K2 K3 but no references were given so that was a dead end.

I'm at work so I can't look it up in my books.
 
Your set is not closed under addition, and thus not a ring.
 
repeat after me ... "I don't know" ... all together now ... "I don't know"...
 
So, you mean the ring *generated* by those four elements?
 
plxmny said:
repeat after me ... "I don't know" ... all together now ... "I don't know"...

I don't know... what you're trying to say, or what your question is if you have one.
 
  • #10
... but name is not so important ... as long as you don't intend to lookup an index ...
 
  • #11
he means multiplication group of course instead of ring
 
  • #12
The last one squared isn't a member of the group. if it was
0 -1
-1 0

then you'd have the klein 4 group (well, isomorphic to it)
 
  • #13
1) it isn't a ring
2) it isn't closed under multiplication.
3) assuming you really mean 'what is the ring generated by R-linear combinations of these elements' (and that you're operating over R), those elements generate all od M_2(R) the 2x2 matrix ring
 

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