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 “” as usually.
It is not difficult to be persuaded of the following:
The sum of a finite amount of ideals is
The sum of any set of ideals consists of all finite sums where every belongs to one of those ideals.
Let in a ring . Because and , we can say that is a of both and .11This may be motivated by the situation in : iff is a factor of . Moreover, is contained in every common factor of and by virtue of its minimality. Hence, may be called the greatest common divisor of the ideals and . The notations
are used, too.
In an analogous way, the intersection of ideals may be designated as the least common of the ideals.
|Title||sum of ideals|
|Date of creation||2013-03-22 14:39:26|
|Last modified on||2013-03-22 14:39:26|
|Last modified by||pahio (2872)|
|Synonym||greatest common divisor of ideals|
|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|