Ernst Kummer invented the concept of ideal numbers to serve as the "missing" factors in number rings in which unique factorization fails; here the word "ideal" is in the sense of … See more

To simplify the description all rings are assumed to be commutative. The non-commutative case is discussed in detail in the respective articles.
Ideals are important … See more

Ideals appear naturally in the study of modules, especially in the form of a radical.
For simplicity, we work with commutative rings but, with … See more

For an arbitrary ring $${\displaystyle (R,+,\cdot )}$$, let $${\displaystyle (R,+)}$$ be its additive group. A subset I is called a left ideal of See more

(For the sake of brevity, some results are stated only for left ideals but are usually also true for right ideals with appropriate notation changes.) See more

The sum and product of ideals are defined as follows. For $${\displaystyle {\mathfrak {a}}}$$ and $${\displaystyle {\mathfrak {b}}}$$ See more

Let A and B be two commutative rings, and let f : A → B be a ring homomorphism. If $${\displaystyle {\mathfrak {a}}}$$ is an ideal in A, then $${\displaystyle f({\mathfrak {a}})}$$ need … See more