A definition of ideal of ring
Topics
About: ring
The table of contents of this article
Starting Context
- The reader knows a definition of ring.
Target Context
- The reader will have a definition of ideal of ring.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Definition
There are 'left ideal', 'right ideal', and 'both-sided ideal'.
left ideal: for any ring, \((S, +, \bullet)\), any additive subgroup, I, of S, such that for any \(p \in S\) and any \(i \in I\), \(p \bullet i \in I\)
right ideal: for any ring, \((S, +, \bullet)\), any additive subgroup, I, of S, such that for any \(p \in S\) and any \(i \in I\), \(i \bullet p \in I\)
both-sided ideal: for any ring, \((S, +, \bullet)\), any additive subgroup, I, of S, such that for any \(p \in S\) and any \(i \in I\), \(p \bullet i \in I\) and \(i \bullet p \in I\)
2: Note
Just 'ideal' is an abbreviation of 'both-sided ideal'.