Fractional ideal

The term fractional ideal is a generalization of the ideal concept from the mathematical field of algebra, which in algebraic number theory in particular plays an important role. In a way, the transition from ordinary to fractional ideals is analogous to the relationship between whole and rational numbers.

This article deals with commutative algebra. In particular, all rings considered are commutative and have an identity element. For more details see Commutative Algebra.

Definition

It should be a Noetherian integral domain and its quotient field.

A fractional ideal is to be a finitely generated submodule of. In some cases it is also required that this not only contains the zero. When you remove this additional condition, so the statement is true, that each (whole ) Ideal particular is a fractional ideal.

A fractional ideal is actually called when the ring

Is the same. ( It is always )

For a broken Ideal Ideal is defined as the inverse

It is a fractional ideal. It is always

Applies equality, so called invertible, and it is

Each fractional principal ideal

For is an invertible fractional ideal. The inverse ideal is

Properties

• A fractional ideal is invertible if there is a projective module.
• Every invertible ideal is actually.
• Is a finite ring extension from. Each fractional ideal is therefore quite finished, so actually is.
• The invertible fractional ideals form a group; their quotient by the subgroup of principal fractional ideals is the ideal class group or Picardgruppe of ( after Charles Emile Picard ).

Examples

• The ideal