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Could someone explain me what the word "copies" in Terms of Rings/Vectorspaces/Modules means? E.g in the context "For a ring R, the smallest subring containing 1 is called the characteristic subring of R. It can be obtained by adding copies of 1 and −1 together many times in any mixture" or
"A free R-module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the ring R. These are the modules that behave very much like vector spaces."

Do copies have special properties? How can one think about them and why is it usefull or easier to work with them instead of the whole ring?

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  • $\begingroup$ In the first example, I wouldn't use the word "copy", though ... there is just one $1$ in every ring, you can't really copy it. $\endgroup$ – Paŭlo Ebermann Jul 2 '18 at 19:24
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"Copies" does not really stand for a mathematical thing. "Direct sum of $n$ copies of $M$" is just the best way English and several indo-european languages have at their disposal to state concisely that a construction such as $X_1\oplus X_2\oplus\cdots \oplus X_n$ (i.e. "direct sum of $n$ objects") is applied to the case $M=X_1=X_2=\cdots=X_n$.

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Part of the point of the word "copies" is that it makes it explicitly clear that the elements of the two copies are different: consider the difference between "the union of $X$ and $X$" and "the union of two copies of $X$". The former just equals $X$ while the latter has cardinality $2|X|$. Then with that in mind, consider "the disjoint union of $X$ and $X$" and "the disjoint union of two copies of $X$". While the former can only ever be interpreted as having cardinality $2|X|$, it's still clearer to use the latter - particularly if you're using a more complicated construction where it isn't obvious that only one interpretation makes sense.

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    $\begingroup$ Put more succinctly, you can pretty safely assume that copies are isomorphic (relative to their context) but not identical. $\endgroup$ – BallBoy Jul 2 '18 at 17:21

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