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I just read the following question Groupification and perfection of a commutative monoid and being also interested in the same topic, I found out the definition of torsion-free monoid I am using is slightly different: (def. 1) a commutative monoid $M$ is torsion-free iff $nx=ny \in M$ for some $n \ge 1$ implies $x=y \in M$. The definition of the linked question consider only the case when $y=0$, call this (def. 2).

Now the two are obviously equivalent when we have additive inverses, i.e. for groups, but for monoids the definition I stated here seems to be strictly stronger: I mean if a monoid is torsion free with def. 1, then it is t.f. for def. 2 but the vice versa is not at all obvious (to me).

Nevertheless, I can't come up with a monoid that verifies def.2 but not def.1, basically because every submonoid of $\mathbb Z^r$ verifies def. 1, while every finite monoid clearly has torsion also for def. 2, so it does not verify the "weak" definition either.

Are the two defs equivalent, or can somebody provide a counterexample?

Moreover, it would be worthwhile to know which of the two definition is "more useful" to think about, when studying commutative monoids algebraic geometry and log schemes.

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up vote 2 down vote accepted

Consider the monoid $M$ obtained by taking $\mathbb N$ (the natural numbers under addition --- including $0$ of course) and then adjoining an element $x$ such that $2x = 2$. Then the elements of $M$ are of the form $n$ or $n + x$ (with $n \in \mathbb N$), and you can check that $M$ is torsion free in the weak sense, but not the strong sense.

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Thanks. Got any idea about the second part of the question, i.e. which definition would prove more useful in studying log schemes? – Niccolò Oct 19 '12 at 4:35

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