# Showing that two rings are not isomorphic

• Mr Davis 97
In summary, to prove that two rings are not isomorphic, one must show that there is no bijective ring homomorphism between them. Isomorphic rings have the same algebraic and structural properties, while non-isomorphic rings have key differences in these properties. Two rings with different underlying sets cannot be isomorphic, and the commutative or non-commutative properties of a ring can affect its isomorphism with another ring. Additionally, two rings with the same number of elements can be non-isomorphic as isomorphism also depends on other properties of the rings.
Mr Davis 97

## Homework Statement

Are the two rings ##R = \mathbb{Z}/3 \times \mathbb{Z}/3## and ##S = (\mathbb{Z}/3)[x]/(x^2+1)## isomorphic or not?

## The Attempt at a Solution

I think that they are not isomorphic. I think this because it seems to be the case that ##(\mathbb{Z}/3)[x]/(x^2+1) \cong (\mathbb{Z}/3)[ i ]##, but that ##(\mathbb{Z}/3)[ i ] \not \cong \mathbb{Z}/3 \times \mathbb{Z}/3##. Am I on the right track or is this wrong?

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Sounds good. You have to explain what ##i## is, namely ##i^2=2## and perhaps you can show what ##\mathbb{Z}_3## is isomorphic to.

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Mr Davis 97

## 1. How do you prove that two rings are not isomorphic?

To prove that two rings are not isomorphic, you need to show that there is no bijective ring homomorphism between the two rings. This can be done by examining their structural properties, such as their orders, subring structures, and ideals.

## 2. What are the key differences between isomorphic and non-isomorphic rings?

Isomorphic rings have the same algebraic properties, such as the same number of elements and operations, and the same structural properties, such as the same number of subrings and ideals. Non-isomorphic rings, on the other hand, have key differences in these properties, making them distinct and not equivalent.

## 3. Can two rings with different underlying sets be isomorphic?

No, two rings with different underlying sets cannot be isomorphic. Isomorphism requires a bijective mapping between the elements of the two rings, which is not possible if the underlying sets are different.

## 4. Do the commutative or non-commutative properties of a ring affect its isomorphism with another ring?

Yes, the commutative or non-commutative properties of a ring can affect its isomorphism with another ring. For example, if one ring is commutative and the other is non-commutative, they cannot be isomorphic since their multiplication operations differ.

## 5. Can two rings with the same number of elements be non-isomorphic?

Yes, two rings with the same number of elements can be non-isomorphic. Isomorphism is not solely dependent on the number of elements, but also on the structural properties and algebraic properties of the rings.

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