PlanetMath: sum of ideals

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sum of ideals

(Definition)

Definition. Let's consider some set of ideals (left, right or two-sided) of a ring. The sum of the 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.

Thus, one can say that the sum ideal is generated by the set of all elements of the individual ideals; in fact it suffices to use all generators of these 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.

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Other names:

greatest common divisor of ideals

Also defines:

sum ideal, sum of the ideals, 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

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Cross-references: algebraic lattice, greatest lower bound, least upper bound, lattice, partially ordered set, intersection, greatest common divisor, contained, factor, iff, divisor, generators, generated by, finite, sum, right, ideals
There are 8 references to this entry.

This is version 17 of sum of ideals, born on 2004-09-29, modified 2008-08-30.
Object id is 6250, canonical name is SumOfIdeals.
Accessed 7290 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)

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