Show one ring not isomorphic to the other

[chycachrrycol]

How would we show that R X R X R X R is not isomorphic to M(R) with R being the set of real numbers.

And more generally, what does it mean for one ring not to be isomorphic to another

 

[Wizlem]

Two rings are not isomorphic if no isomorphism exists between them. One way to show that the isomorphism doesn't exist is to assume it exists and then find elements which do not satisfy the definition of isomorphism to contradict the assumption.

M(R) is the set of matrices with real elements? If so, you could use the fact that M(R) is non-commutative or any other difference between R and M(R) to show they aren't isomorphic.