2022-06-26

309: Ideal of Ring

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A definition of ideal of ring

Topics


About: ring

The table of contents of this article


Starting Context



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'.


References


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