PlanetMath: sum of ideals

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (203)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

sum of ideals

(Definition)

Definition. Let's consider some set of ideals (left, right or two-sided) of a ring. The sum of ideals is the smallest ideal of the ring containing all those ideals. The sum of ideals is denoted by using “+” and “ $\sum$ ” as usually.

It is not difficult to be persuaded of the following:

The sum of a finite amount of ideals is
$\displaystyle \mathfrak{a}_1+\mathfrak{a}_2+\cdots+\mathfrak{a}_k = \{a_1\!+\!a_2\!+\!\cdots\!+\!a_k\,\vdots \quad a_i \in \mathfrak{a}_i \,\,\forall i\}.$
The sum of any set of ideals consists of all finite sums $\displaystyle\sum_j a_j$ where every belongs to one $\mathfrak{a}_j$ of those ideals.

Let $\mathfrak{a}+\mathfrak{b} = \mathfrak{d}$ in a ring . Because $\mathfrak{a} \subseteq \mathfrak{d}$ and $\mathfrak{b} \subseteq \mathfrak{d}$ , we can say that $\mathfrak{d}$ is a factor or divisor of both $\mathfrak{a}$ and $\mathfrak{b}$ .¹ Moreover, $\mathfrak{d}$ is contained in every common factor $\mathfrak{c}$ of $\mathfrak{a}$ and $\mathfrak{b}$ by virtue of its minimality. Hence, $\mathfrak{d}$ may be called the greatest common divisor of the ideals $\mathfrak{a}$ and $\mathfrak{b}$ . The notations

$\displaystyle \mathfrak{a}+\mathfrak{b} = \gcd(\mathfrak{a}, \,\mathfrak{b}) = (\mathfrak{a}, \,\mathfrak{b})$

are used, too.

In an analogous way, the intersection of ideals may be designated as the least common multiple of the ideals.

The by “ $\subseteq$ ” partially ordered set of all ideals of a ring forms a lattice, where the least upper bound of $\mathfrak{a}$ and $\mathfrak{b}$ is $\mathfrak{a+b}$ and the greatest lower bound is $\mathfrak{a\cap b}$ . See also the example 3 in algebraic lattice.

Footnotes

....¹: This may be motivated by the situation in $\mathbb{Z}$ : $(n) \subseteq (m)$ iff is a factor of .

"sum of ideals" is owned by pahio.

(view preamble)

Other names:

greatest common divisor of ideals

Also defines:

sum ideal, addition of ideals, factor of ideal, greatest common divisor of ideals, least common multiple of ideals

This object's parent.

Attachments:: multiplication ring (Definition) by PrimeFan

Log in to rate this entry.
(view current ratings)

Cross-references: algebraic lattice, greatest lower bound, least upper bound, lattice, partially ordered set, intersection, greatest common divisor, contained, iff, divisor, factor, finite, sum, right, ideals
There are 7 references to this entry.

This is version 16 of sum of ideals, born on 2004-09-29, modified 2008-04-16.
Object id is 6250, canonical name is SumOfIdeals.
Accessed 6712 times total.

Classification:

AMS MSC:	08A99 (General algebraic systems :: Algebraic structures :: Miscellaneous)
	16D25 (Associative rings and algebras :: Modules, bimodules and ideals :: Ideals)
	13C99 (Commutative rings and algebras :: Theory of modules and ideals :: Miscellaneous)

Pending Errata and Addenda

None.[ View all 1 ]

Discussion

forum policy

No messages.

Interact