In previous post, we’ve discussed about “Rings“. Now, this post talks about what are Ideals, and its definitions that uses base of rings and theorems on the topic..
Let us first define Ideals-
What are Ideals ?
In order to understand Ideal, we can connect it with the concept of ‘Coset’ in algebra. It is also a coset just the difference is this is coset over a ring R. Let us look at definition-
A non empty subset U of some ring R is an ideal of R if it satisfy following criteria-
- U must be a subgroup of a ring R under operation addition.
- For every u ∈ U & r ∈ R, both ur and ru are in U.
There is also another definition, that links to the concept ‘Normal subgroup’. Let us look at it..
A nonempty subset S of a of a ring R is an Ideal of R if it satisfies-
- a, b ∈ S implies a-b ∈ S.
- ar ∈ S (ra ∈ S) for all a ∈ S and r ∈ R.
Every ring has two ideals these are- (0) and R itself. These are Trivial ideals.
If as in definition, r ∈ R is at the left side (ra ∈ S) is a left ideal, whereas, if r is at right side i.e.(ar ∈ S) is a right ideal.
Sometimes an ideal is two sided then we call it simply ideal.
We consider an example in order to get a clear idea–
Consider R be the n×n matrix ring over field F, for any 1≤ i ≤n let A be the set of matrices in a ring R having all rows , except possibly the ith, zero. Then A is right ideal.
This is about What are ideals & their definitions. In further post we’ll discuss about their types…