Motivation
Intuitively, the definition can be motivated as follows: Suppose we have a subset of elements Z of a ring R and that we would like to obtain a ring with the same structure as R, except that the elements of Z should be zero (they are in some sense "negligible").
But if and in our new ring, then surely should be zero too, and as well as should be zero for any element r (zero or not).
The definition of an ideal is such that the ideal I generated (see below) by Z is exactly the set of elements that are forced to become zero if Z becomes zero, and the quotient ring R/I is the desired ring where Z is zero, and only elements that are forced by Z to be zero are zero. The requirement that R and R/I should have the same structure (except that I becomes zero) is formalized by the condition that the projection from R to R/I is a (surjective) ring homomorphism.
Read more about this topic: Ideal (ring Theory)
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