M be a module over a ring R and S be a multiplicatively closed subset of R. An element m of M is called an S-torsion element if there exists an element s in S such that s annihilates m, i.e., In particular, one can take for S the set of regular elements of the ring R and recover the definition above. An element g of a group G is called a torsion element of the group if it has finite order, i.e., if there is a positive integer m such that gm = e, where e denotes the identity element of the group, and gm denotes the product of m copies of g. A group is called a torsion (or periodic) group if all its elements are torsion elements, and a if its only torsion element is the identity element. Any abelian group may be viewed as a module over the ring Z of integers, and in this case the two notions of torsion coincide. Examples Let M be a free module over any ring R. Then it follows immediately from the definitions that M is torsion-free (if the ring R is not a domain then torsion is considered with respect to the set S of non-zero-divisors of R). In particular, any free abelian group is torsion-free and any vector space over a field K is torsion-free when viewed as the module over K. By contrast with example 1, any finite group (abelian or not) is periodic and finitely generated. Burnside's problem, conversely, asks whether any finitely generated periodic group must be finite? The answer is "no" in general, even if the period is fixed. The torsion elements of the multiplicative group of a field are its roots of unity. In the modular group, Γ obtained from the group SL(2, Z) of 2×2 integer matrices with unit determinant by factoring out its center, any nontrivial torsion element either has order two and is conjugate to the element S or has order three and is conjugate to the element ST. In this case, torsion elements do not form a subgroup, for example, S·ST = T, which has infinite order. The abelian group Q/Z, consisting of the rational numbers modulo 1, is periodic, i.e. every element has finite order. Analogously, the module K(t)/K[t] over the ring R = K[t] of polynomials in one variable is pure torsion. Both these examples can