1. ## Isomorphisms between rings

Hello.
Don't understand these isomorphism definitions in one of my mathematics classes.
Help appreciated.

Prove that no two of the following rings are isomorphic:
(i) R x R x R x R (with addition and multiplication given coordinate by coordinate)
(ii) A two by two matrix in R
(iii) The ring of real quaternions (H)

Thank you.
Timothy

2. Originally Posted by THulchenko
Hello.
Don't understand these isomorphism definitions in one of my mathematics classes.
Help appreciated.

Prove that no two of the following rings are isomorphic:
(i) R x R x R x R (with addition and multiplication given coordinate by coordinate)
(ii) A two by two matrix in R
(iii) The ring of real quaternions (H)
Isomorphism, informally, means identical structure. The way to show two things are not isomorphic is by showing one has a property another does not.
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(i) and (iii) the ring of quaternions is a first example of a strictly skew field. That means the quaternions are not commutative ring. While R x R x R x R is for a direct product of commutative rings.

(i) and (ii) same idea. Matrix multiplication is not commutative.

(ii) and (iii), well, the ring of matrices are not a strictly skew field. In more basic terms, the ring of matrices is not a division ring (not non-zero everything has a multiplicative inverse) while the quaternions are a division ring (everything non-zero has a multiplicative inverse).