In mathematics, specifically in the area of abstract algebra, an ideal is a subset of a ring that is closed under the ring's operations of addition and multiplication by elements of the ring. Ideals generalize the concept of a principal ideal, which is a subset generated by a single element of the ring. Here are some key points about ideals:
Definition
Given a ring ( R ) and a subset ( I ) of ( R ), ( I ) is an ideal if the following conditions hold:
1. Closure under addition: For all ( a, b in I ), ( a + b in I ).
2. Closure under multiplication by ring elements: For all ( a in I ) and ( r in R ), ( ar in I ) and ( ra in I ).
Types of Ideals
Principal ideals: These are ideals generated by a single element, i.e., ( (a) = {ra + sa : r, s in R
