Is ℤ3 a Subring of ℤ? Exploring the Relationship and Debunking Myths

  • Thread starter Bachelier
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In summary, a subring is a subset of a ring that is closed under addition, subtraction, and multiplication and contains the additive identity and multiplicative identity of the original ring. ℤ3 is a subring of ℤ because it meets the criteria of being a subset of the ring ℤ and is closed under addition, subtraction, and multiplication. A subgroup is a subset of a group that is closed under the group operation and contains the identity element. The main difference between a subgroup and a subring is that a subgroup is a subset of a group, while a subring is a subset of a ring. A subring can have more elements than the original ring as long as it meets the criteria of being a subset and being closed
  • #1
Bachelier
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By the way ℤ3 (or ℤ/3ℤ) (mod 3) is not a subring of ℤ, is it?

thanks
 
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  • #2
Bachelier said:
By the way ℤ3 (or ℤ/3ℤ) (mod 3) is not a subring of ℤ, is it?

thanks

no. Z3 is torsion, Z is free. Z3 is a quotient ring of Z by 3Z.
 
  • #3
These are different types of objects, Bachelier; Z/3 is a set of equivalence classes, and Z is a collection of numbers. If maybe you mean whether Z/3 can be embedded in Z as a subring, the answer is no, by, e.g., Lavinia's argument.
 

What is a subring?

A subring is a subset of a ring that is closed under addition, subtraction, and multiplication and contains the additive identity and multiplicative identity of the original ring.

Is ℤ3 a subring of ℤ?

Yes, ℤ3 is a subring of ℤ because it meets the criteria of being a subset of the ring ℤ and is closed under addition, subtraction, and multiplication. It also contains the additive identity (0) and multiplicative identity (1) of ℤ.

How is a subring different from a subgroup?

A subgroup is a subset of a group that is closed under the group operation and contains the identity element. A subring is a subset of a ring that is closed under addition, subtraction, and multiplication and contains the additive identity and multiplicative identity. The main difference is that a subgroup is a subset of a group, while a subring is a subset of a ring.

Can a subring have more elements than the original ring?

Yes, a subring can have more elements than the original ring. This is because a subring only needs to be a subset of the original ring and meet the criteria of being closed under addition, subtraction, and multiplication and containing the additive identity and multiplicative identity. It does not have to have the same number of elements as the original ring.

What is the importance of determining if ℤ3 is a subring of ℤ?

Determining if ℤ3 is a subring of ℤ can help in understanding the relationship between the two mathematical structures and how they interact. It also allows for the application of ring theory and its properties to ℤ3, which can provide insights and solutions to various mathematical problems.

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