# Isomorphic Ring

• Oct 31st 2008, 05:35 AM
vincisonfire
Isomorphic Ring
Prove that no two of the following rings are isomorphic:
(a) R × R × R × R (with addition and multiplication given coordinate by coordinate);
(b) M2 (R) (2 x 2 matrices with real coefficient);
(c) The ring H of real quaternions (are they simply integers?).
• Oct 31st 2008, 05:51 AM
ThePerfectHacker
Quote:

Originally Posted by vincisonfire
(a) R × R × R × R (with addition and multiplication given coordinate by coordinate);
(b) M2 (R) (2 x 2 matrices with real coefficient);
(c) The ring H of real quaternions (are they simply integers?).

1) Ring (a) is commutive but ring (b) and (c) are not.
2) Ring (c) is a strictly skew field but (b) is not.
• Oct 31st 2008, 06:37 AM
vincisonfire
For number two you mean :
- There exists an inverse with respect to multiplication for ALL quaternions.
- There exists an inverse with respect to multiplication for SOME real 2x2 matrices.
That's it?
• Oct 31st 2008, 10:52 AM
ThePerfectHacker
Quote:

Originally Posted by vincisonfire
For number two you mean :
- There exists an inverse with respect to multiplication for ALL quaternions.
- There exists an inverse with respect to multiplication for SOME real 2x2 matrices.
That's it?

Exactly. In $M_{2\times 2}(\mathbb{R})$ we have that not all non-zero elements have inverse. Whiles with $\mathbb{H}$ (quaternions) all non-zero elements have inverses.