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

(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.)
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The sum and product of ideals are defined as follows. For $${\displaystyle {\mathfrak {a}}}$$ and $${\displaystyle {\mathfrak {b}}}$$, left (resp. right) ideals of a ring R, their sum is 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 $${\displaystyle R}$$ if it is an additive subgroup 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 … 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 $${\displaystyle f({\mathfrak {a}})}$$ See more