Prove Isomorphism: R x S & S x R

• stripes
In summary, the conversation discusses the task of showing that for any rings R and S, R x S and S x R are isomorphic, with R x S being the cartesian product or ordered pairs. The focus is on finding a bijection that preserves addition and multiplication, with the suggestion to use a "natural" mapping.

Homework Statement

Show that for any rings R and S, R x S and S x R are isomorphic, and A x B is the cartesian product, or ordered pairs. So an element of R x S can be written as (r1, s1).

The Attempt at a Solution

So I have to show that there is a bijection from R x S to S x R, and this bijection must preserve addition and multiplication. This is tough for me since the mapping from R x S to S x R could be anything! How can I even start if I don't have this function or mapping?

hi stripes!
stripes said:
… the mapping from R x S to S x R could be anything!

yes it could

but you're in charge, and you can choose any mapping you like

go for the "natural" mapping …

what do you think it would be really neat for (r1, s1) to be mapped onto?

What is "Prove Isomorphism: R x S & S x R"?

"Prove Isomorphism: R x S & S x R" is a mathematical concept that involves showing that two sets, R x S and S x R, are isomorphic, or structurally identical. This means that there exists a bijective function between the two sets that preserves their structure.

What does it mean for two sets to be isomorphic?

Two sets are isomorphic if there exists a bijective function between them that preserves their structure. This means that the two sets have the same number of elements and the same relationships between those elements.

How do you prove isomorphism between two sets?

To prove isomorphism between two sets, you must show that there exists a bijective function between the two sets that preserves their structure. This can be done by explicitly defining the function and showing that it is both one-to-one and onto.

What are some common examples of isomorphic sets?

Some common examples of isomorphic sets include the set of real numbers and the set of complex numbers, the set of even integers and the set of odd integers, and the set of rational numbers and the set of integers.

Why is proving isomorphism important in mathematics?

Proving isomorphism is important in mathematics because it allows us to understand the relationships between different sets and see that they have the same underlying structure. This can help us solve problems and make connections between seemingly unrelated concepts.