We can define some additional arithmetic generalizations for rings:
- Ring unity 1 (AKA identity): identity under multiplication; a ring with unity is unital (AKA unitary)
- Ring unit (AKA invertible element): nonzero element a of unital ring with multiplicative inverse aa−1=a−1a=1
- Idempotent element: element a such that a2=a
- Nilpotent element: there exists an integer n such that an=0
- Ring characteristic: the least n∈Z+ such that na=0∀a∈R; 0 if n does not exist
| Δ It is important to remember that a ring may not have an identity (unity) or inverses under multiplication. However, it should also be noted that “ring” is sometimes defined to include a unity, in which case a ring without unity is called a rng (“ring without the i”). |
As higher structure is added to a ring, it begins to severely constrain its form:
- Every integral domain has characteristic 0 or prime
- Every finite integral domain is a field
- Every finite field (AKA Galois field) has order pn with p prime (denoted GF(pn) or Fpn), and is unique (up to isomorphism)
- GF(p) is isomorphic to Zp, the integers modulo p (also denoted Z/pZ or Z/(p))
We do not discuss ideals here, which are to rings as normal subgroups are to groups, and so are also covered in Dividing algebraic objects.