Broken ideal

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

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. For more details, see Commutative Algebra .

definition

Let it be a Noetherian integrity ring and its quotient body . ${\displaystyle A}$ ${\displaystyle K}$

A broken ideal zu is a finitely generated - sub-module of . Sometimes it is also required that this does not only contain the zero. If one waives this additional condition, then the statement applies that every (whole) ideal is in particular a broken ideal. ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle K}$

A broken ideal is actually called when the ring ${\displaystyle {\mathfrak {a}}}$

${\displaystyle \mathrm {End} \,{\mathfrak {a}}=\{x\in K\mid x{\mathfrak {a}}\subseteq {\mathfrak {a}}\}}$

is the same . (It always applies ) ${\displaystyle A}$ ${\displaystyle A\subseteq \mathrm {End} \,{\mathfrak {a}}.}$

To a broken ideal , the inverse ideal is defined as ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle {\mathfrak {a}}^{-1}}$

${\displaystyle {\mathfrak {a}}^{-1}=\{x\in K\mid x{\mathfrak {a}}\subseteq A\}.}$

It's a broken ideal. It always applies

${\displaystyle {\mathfrak {a}}{\mathfrak {a}}^{-1}\subseteq A.}$

If equality holds, it is called invertible , and it is ${\displaystyle {\mathfrak {a}}}$

${\displaystyle {\mathfrak {a}}=({\mathfrak {a}}^{-1})^{-1}.}$

Every broken main ideal

${\displaystyle (a)=A\cdot a=\{x\cdot a\mid x\in A\}}$

for is an invertible broken ideal. The inverse ideal is ${\displaystyle a\in K^{\times }}$ ${\displaystyle (a^{-1}).}$

properties

• A broken ideal is invertible if and only if it is a projective module. ${\displaystyle A}$
• Every invertible ideal is real.
• ${\displaystyle \mathrm {End} \,{\mathfrak {a}}}$is a finite ring expansion of . So if it is completely closed , then every broken ideal is actually. ${\displaystyle A}$ ${\displaystyle A}$
• The invertible broken ideals form a group; their quotient after the subgroup of the broken main ideals is the ideal class group or Picard group of (after Charles Emile Picard ). ${\displaystyle \mathrm {Pic} \,A}$ ${\displaystyle A}$

Examples

• The ideal

${\displaystyle {\mathfrak {a}}=(2,1+{\sqrt {5}})\subseteq \mathbb {Z} [{\sqrt {5}}]}$

is not actually because

${\displaystyle \mathrm {End} \,{\mathfrak {a}}=\mathbb {Z} \!\left[{\frac {1+{\sqrt {5}}}{2}}\right].}$

See also

• Dedekindring