Ideal

ideal, in modern algebra, a subring of a mathematical ring with certain absorption properties. The concept of an ideal was first defined and developed by German mathematician Richard Dedekind in 1871. In particular, he used ideals to translate ordinary properties of arithmetic into properties of sets.

A ring is a set having two binary operations, typically addition and multiplication. Addition (or another operation) must be commutative (a + b = b + a for any a, b) and associative [a + (b + c) = (a + b) + c for any a, b, c], and multiplication (or another operation) must be associative [a(bc) = (ab)c for any a, b, c]. There must also be a zero (which functions as an identity element for addition), negatives of all elements (so that adding a number and its negative produces the ring’s zero element), and two distributive laws relating addition and multiplication [a(b + c) = ab + ac and (a + b)c = ac + bc for any a, b, c]. A subset of a ring that forms a ring with respect to the operations of the ring is known as a subring.

For a subring I of a ring R to be an ideal, ax and xa must be in I for all a in R and x in I. In other words, multiplying (on the left or right) any element of the ring by an element of the ideal produces another element of the ideal. Note that ax may not equal xa, as multiplication does not have to be commutative.

Furthermore, each element a of R forms a coset (a + I), where every element from I is substituted into the expression to produce the full coset. For an ideal I, the set of all cosets forms a ring, with addition and multiplication, respectively, defined by: (a + I) + (b + I) = (a + b) + I and (a + I)(b + I) = ab + I. The ring of cosets is called a quotient ring R/I, and the ideal I is its zero element. For example, the set of integers (ℤ) forms a ring with ordinary addition and multiplication. The set 3ℤ formed by multiplying each integer by 3 forms an ideal, and the quotient ring ℤ/3ℤ has only three elements:

1. 0 + 3ℤ = 3ℤ = {0, ±3, ±6, ±9,…}
2. 1 + 3ℤ = {…, −8, −5, −2, 1, 4, 7,…}
3. 2 + 3ℤ = {…, −7, −4, −1, 2, 5, 8,…}