## What is torsion element?

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In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. A torsion module is a module that equals its torsion submodule.

## What is a torsion point?

Definition 3 A point P ∈ C(Q) is called a torsion point of order n if P has order n. Gathering all of the torsion points of a an elliptic curve C will form a finite subgroup of C(Q), called C(Q)tor: C(Q)tor = {P ∈ C(Q) | P has finite order} ⊆ C(Q).

**What does it mean for a group to be torsion free?**

a group in which every element other than the identity has infinite order.

**Is a torsion free?**

In algebra, a torsion-free module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In other words, a module is torsion free if its torsion submodule is reduced to its zero element.

### What is a torsion R module?

A torsion module is a module whose elements are all torsion. A torsion-free module is a module whose elements are not torsion, other than 0. More generally, given an ideal 𝔞⊂R then an 𝔞-torsion module is one all whose elements are annihilated by some power of elements in 𝔞.

### What is the rank of a module?

The rank of a free module M over an arbitrary ring R( cf. Free module) is defined as the number of its free generators. For rings that can be imbedded into skew-fields this definition coincides with that in 1). In general, the rank of a free module is not uniquely defined.

**How do torsion springs work?**

A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy when it is twisted. When it is twisted, it exerts a torque in the opposite direction, proportional to the amount (angle) it is twisted.

**Which group is torsion-free group?**

free abelian group

In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only element with finite order.

## What is torsion in differential geometry?

In the differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting out of the osculating plane. Taken together, the curvature and the torsion of a space curve are analogous to the curvature of a plane curve.

## What is free Z module?

A finitely generated torsion-free module of a commutative PID is free. A finitely generated Z-module is free if and only if it is flat. See local ring, perfect ring and Dedekind ring.