rigRig2004-10-16 12:02:472006-09-15 03:25:25Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA \emph{rig} $(R, +, \cdot)$ is a set $R$ together with two binary operations $+:R^2 \to R:(a, b) \mapsto a + b$ and $\cdot:R^2 \to R:(a, b) \mapsto ab$, such that both $(R, +)$ and $(R, \cdot)$ are monoids, where $\cdot$ distributes over $+$. That is if $\{a, b, c, d\} \subset R$ then $(a+b)(c+d) = ac + ad + bc +bd$. The natural numbers with ordinary addition and multiplication $(\mathbf{N}, +, \cdot)$ is a rig.
A rig $(R, +, \cdot)$ is a ring if $(R, +)$ is a group. The integers with ordinary addition and multiplication $(\mathbf{Z}, +, \cdot)$ is a ring.
%A ring $(R, +, \cdot)$ is a field if both $(R, +)$ and $(R, \cdot)$ are groups. The fractions $(\mathbf{Q}, +, \cdot)$, reals $(\mathbf{R}, +, \cdot)$, complex numbers $(\mathbf{C}, +, \cdot)$ and quaternions $(\mathbf{H}, +, \cdot)$ with ordinary addition and multiplication are fields.