By the way ℤ3 (or ℤ/3ℤ) (mod 3) is not a subring of ℤ, is it?
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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