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

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SUMMARY

The set of units in a ring R with identity is not a subring of R. This conclusion is drawn from the fact that the zero element (0R) cannot be a unit, as it does not have a multiplicative inverse. Therefore, the set of units fails to satisfy the necessary conditions for being a subring, specifically the requirement of containing the additive identity. The discussion emphasizes the importance of closure under addition and multiplication for subgroup verification.

PREREQUISITES
  • Understanding of ring theory and the definition of a ring with identity.
  • Familiarity with the concept of units in a ring.
  • Knowledge of subgroup criteria, including closure under addition and multiplication.
  • Basic proof techniques in abstract algebra.
NEXT STEPS
  • Study the properties of rings and their identities in more depth.
  • Learn about the structure of subrings and the conditions required for a subset to be a subring.
  • Explore examples of units in various rings, such as integers and polynomial rings.
  • Investigate the implications of closure properties in group theory and ring theory.
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Students of abstract algebra, mathematicians exploring ring theory, and anyone interested in the properties of algebraic structures.

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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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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?
 

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