# Proving Two Rings Are Isomorphic

• MHB
• cs0978
In summary, the problem asks to prove that the rings R and S are isomorphic, and the first step is to find the zero and unit elements in each ring.
cs0978
I seem to be having a difficult time trying to figure out how to do this problem. It's from a non-graded homework assignment. I was able to get every other problem except for this one.Let R = Z, together with the two operations:
a + b := a + b + 3 and ab := ab + 3a + 3b + 6

Let S = Z, together with the two operations:
a + b := a + b - 2 and ab := -ab + 2a + 2b - 2

Assume ordinary addition and multiplication in Z and that R and S are rings.

Prove that R is isomorphic to S.

cs0978 said:
I seem to be having a difficult time trying to figure out how to do this problem. It's from a non-graded homework assignment. I was able to get every other problem except for this one.Let R = Z, together with the two operations:
a + b := a + b + 3 and ab := ab + 3a + 3b + 6

Let S = Z, together with the two operations:
a + b := a + b - 2 and ab := -ab + 2a + 2b - 2

Assume ordinary addition and multiplication in Z and that R and S are rings.

Prove that R is isomorphic to S.
As a start, find what the zero and unit elements are in $R$ and $S$. An isomorphism must take the zero element of $R$ to the zero element of $S$, and the same for the unit elements.

## 1. What does it mean for two rings to be isomorphic?

Two rings are isomorphic if there exists a bijective map between them that preserves the ring structure. This means that the two rings have the same number of elements and the same operations (addition and multiplication) with the same properties.

## 2. How do you prove that two rings are isomorphic?

To prove that two rings are isomorphic, you need to show that there exists a bijective map between them, and that this map preserves the ring structure. This can be done by defining a map between the two rings and showing that it is both one-to-one and onto, as well as proving that it preserves the ring operations.

## 3. Can two rings be isomorphic if they have different elements?

No, two rings must have the same number of elements in order to be isomorphic. If two rings have different elements, there is no bijective map that can preserve the ring structure between them.

## 4. Is it possible for two non-commutative rings to be isomorphic?

Yes, two non-commutative rings can still be isomorphic as long as there exists a bijective map that preserves the ring structure. The commutativity of the ring does not affect its isomorphism.

## 5. Are there any other properties that must be preserved for two rings to be isomorphic?

In addition to the ring operations, the identity elements and the distributive property must also be preserved for two rings to be isomorphic. This means that the identity elements in one ring must map to the identity elements in the other ring, and the distributive property must hold for both rings under the given map.

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