Let's look at more operations. Consider an analog of addition, +, and an analog of multiplication, *, acting on a set. This is a "ring" if it satisfies these conditions:
- Addition forms an abelian group (identity 0, inverse of a: -a)
- Multiplication is associative
- Multiplication has an identity (1)
- Multiplication is distributive over addition
Some mathematicians define a ring as not necessarily having a multiplicative identity. Relative to a ring with one, such a ring is sometimes called a rng ("rung"; no i) or a pseudoring. But I'll use the multiplicative identity here.
The additive identity, 0, is the multiplicative zero.
A ring with 0 = 1 is the "zero ring", with only one element.
Every ring contains either the integer ring, Z, or else the ring of integers modulo some number n, Z(n). This subring commutes with all the other elements in the ring.
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Wikipedia lists a chain of ring-like objects starting with commutative rings and ending with algebraic fields.
- Commutative rings: multiplication is commutative.
- Integral domain: no zero divisors. a*b = 0 implies a = 0 or b = 0. An integral domain has the "cancellation property": a*b = a*c implies b = c for all a. It's easy to prove that this follows from the lack of zero divisors.
- Integrally closed domains -- rather complicated.
- A GCD domain has the property that any two elements have a nonzero greatest common divisor: a and b have g such that a = g*a1 and b = g*b1.
- A unique factorization domain has the property that every element other than 0 can be written as a unique product of a unit element (1, -1, etc.) and some "prime elements". Here is a ring that is not a UFD: the ring of numbers with the form a + b*sqrt(-5), where a and b are integers. 6 = 2*3 = (1 + sqrt(-5))*(1 - sqrt(-5)).
- A principal ideal domain has the property that every ideal is principal, generated by only one element. A left ideal of a ring: (ideal)*(ring) = (ideal). A right ideal: (ring)*(ideal) = (ideal). If both sides, then a two-sided ideal. An ideal of a commutative ring is always a two-sided ideal. The integer ring Z has ideals n*Z, generated by n.
- A Euclidean domain has a generalization of Euclid's GCD algorithm in it.
- An algebraic field has the property that multiplication over all elements but 0 is an abelian group.
Though the integers do not form a field, the rational numbers, the algebraic numbers, the real numbers, and various other sorts of numbers are all fields.
A finite field is a field with a finite number of elements. Sometimes called Galois fields, after the famous mathematician Évariste Galois (no, he didn't write down all his mathematical discoveries the night before his fatal duel).