What is torsion element?

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.