convention. rings and fields [gathmann2013commutative, 0.1] [ag-0008]

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A ring, usually denoted \(R\), is always assumed to be a commutative ring (i.e. \(a+b=b+a\) and \(a b=b a\) for all \( a, b \in R\)) with 1 (i.e. the multiplicative identity element, or called multiplicative unit).

\(1 \neq 0\) is not required, where \(0\) is the additive neutral element. If \(1=0\), then \(R\) must be the zero ring (or called the trivial ring), which consisting of one element, and is denoted \(\{0\}\).

Subrings must have the same unit, and ring homomorphisms are always required to map \(1\) to \(1\).

A field, usually denoted \(K\), is a commutative ring with \(1\), where every nonzero element has a multiplicative inverse (thus division can be defined).