sum of ideals

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

 $\mathfrak{a}_{1}+\mathfrak{a}_{2}+\cdots+\mathfrak{a}_{k}\;=\;\{a_{1}\!+\!a_{2% }\!+\!\cdots\!+\!a_{k}\,\vdots\quad a_{i}\in\mathfrak{a}_{i}\,\,\forall i\}.$
• The sum of any set of ideals consists of all finite sums $\displaystyle\sum_{j}a_{j}$ where every $a_{j}$ 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 $R$.  Because  $\mathfrak{a}\subseteq\mathfrak{d}$  and  $\mathfrak{b}\subseteq\mathfrak{d}$,  we can say that $\mathfrak{d}$ is a of both $\mathfrak{a}$ and $\mathfrak{b}$.11This may be motivated by the situation in $\mathbb{Z}$:  $(n)\subseteq(m)$  iff  $m$ is a factor of $n$.  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 of the ideals $\mathfrak{a}$ and $\mathfrak{b}$.  The notations

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

are used, too.

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

 Title sum of ideals Canonical name SumOfIdeals Date of creation 2013-03-22 14:39:26 Last modified on 2013-03-22 14:39:26 Owner pahio (2872) Last modified by pahio (2872) Numerical id 22 Author pahio (2872) Entry type Definition Classification msc 13C99 Classification msc 16D25 Classification msc 08A99 Synonym greatest common divisor of ideals Related topic QuotientOfIdeals Related topic ProductOfIdeals Related topic LeastCommonMultiple Related topic TwoGeneratorProperty Related topic Submodule Related topic AlgebraicLattice Related topic LatticeOfIdeals Related topic MaximalIdealIsPrime Related topic AnyDivisorIsGcdOfTwoPrincipalDivisors Related topic GcdDomain Defines sum ideal Defines sum of the ideals Defines addition of ideals Defines factor of ideal Defines greatest common divisor of ideals Defines least common multiple of ideals
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