# Rings

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.