Is ℤ3 a Subring of ℤ? Understanding the Mathematical Relationship

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SUMMARY

ℤ3 (or ℤ/3ℤ) is definitively not a subring of ℤ. This conclusion is based on the fact that ℤ3 is a torsion ring, while ℤ is a free ring. ℤ3 is classified as a quotient ring of ℤ by 3ℤ, representing a set of equivalence classes, in contrast to ℤ, which is a collection of integers. The inability to embed ℤ3 in ℤ as a subring is supported by Lavinia's argument.

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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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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.
 
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.
 

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