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

[plxmny]

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

 

[NateTG]

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

 

[plxmny]

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.

 

[NateTG]

If it's a ring, where's zero, and what are the addition and multiplication operations?

 

[plxmny]

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.

 

[DeadWolfe]

Your set is not closed under addition, and thus not a ring.

 

[plxmny]

repeat after me ... "I don't know" ... all together now ... "I don't know"...

 

[NateTG]

So, you mean the ring *generated* by those four elements?

 

[DeadWolfe]

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.

 

[lalbatros]

... but name is not so important ... as long as you don't intend to lookup an index ...

 

[Ultraworld]

he means multiplication group of course instead of ring

 

[Office_Shredder]

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)

 

[matt grime]

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