Is the Set of Units in a Ring with Identity a Subring?

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In summary, the conversation discusses whether the set of units in a ring R with identity is a subring of R. The attempted solution presents a proof that disproves this claim by showing that the set of units does not contain the additive identity. However, the other participant suggests that the identity being referred to is the additive one, and proposes a different approach to disprove the claim by checking if the set is closed under addition.
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Homework Statement



Prove or Disprove: The set of units in a ring R with identiy is a subring of R.


Homework Equations





The Attempt at a Solution



Let S be the the set of units in a ring R with identity. For S to be a subring of R, 0R would have to be an element of S. Since S is the set of units in R, it follows that S will not a multiplicative identity, namely 0R*0R-1 is not an element of S. Hence S is not a subring of R, disproving the original claim.


I feel that the fact 0R*0R-1 is not an element of S is the main part of the proof. I am just unsure if my argument and logic are correct.
 
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  • #2
I think the identity he is referring to is the additive one (1 is trivially a unit). So your counter proof isn't really valid.

If S was to be a subgroup then it must be closed under addition and multiplication. It is easy to check that its closed under multiplication. Look at addition, when you add two units, is it always a unit?
 

1. What is a subring?

A subring is a subset of a ring that is itself a ring, meaning it has all the same operations and properties as the original ring.

2. How do you prove that a subset is a subring?

To prove that a subset is a subring, you must show that it satisfies the following conditions: it is closed under addition and multiplication, it contains the additive and multiplicative identities of the original ring, and it is closed under additive and multiplicative inverses.

3. Can a subring be a proper subset of the original ring?

Yes, a subring can be a proper subset of the original ring. This means that the subring does not contain all the elements of the original ring, but it still satisfies the conditions to be considered a subring.

4. How do you disprove that a subset is a subring?

To disprove that a subset is a subring, you can provide a counterexample where one of the conditions for a subring is not satisfied. For example, if the subset is not closed under addition, it cannot be a subring.

5. Can a subring have a different set of operations than the original ring?

No, a subring must have the same set of operations as the original ring in order to be considered a subring. This includes the same addition, multiplication, and negation operations.

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