Definition 9.2.1. A *field* is a nonzero ring where every nonzero element is invertible. Given a field a *subfield* is a subring that is itself a field.

## 9.2 Basic definitions

Because we have placed this chapter before the chapter discussing commutative algebra we need to introduce some of the basic definitions here before we discuss these in greater detail in the algebra chapters.

For a field $k$, we write $k^*$ for the subset $k \setminus \{ 0\} $. This generalizes the usual notation $R^*$ that refers to the group of invertible elements in a ring $R$.

Definition 9.2.2. A *domain* or an *integral domain* is a nonzero ring where $0$ is the only zerodivisor.

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