We can define some additional arithmetic generalizations for rings:

  • Ring unity \({\mathbf{1}}\) (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 \({aa^{-1}=a^{-1}a=\mathbf{1}}\)
  • Idempotent element: element \({a}\) such that \({a^{2}=a}\)
  • Nilpotent element: there exists an integer \({n}\) such that \({a^{n}=\mathbf{0}}\)
  • Ring characteristic: the least \({n\in\mathbb{Z}^{+}}\) such that \({na=\mathbf{0}\;\forall a\in R}\); 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 \({p^{n}}\) with \({p}\) prime (denoted \({GF(p^{n})}\) or \({\mathbb{F}_{p^{n}}}\)), and is unique (up to isomorphism)
  • \({GF(p)}\) is isomorphic to \({\mathbb{Z}_{p}}\), the integers modulo \({p}\) (also denoted \({\mathbb{Z}/p\mathbb{Z}}\) or \({\mathbb{Z}/\left(p\right)}\))

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.

An Illustrated Handbook