nLab characteristic zero

Definition

Every ring $R$ has a characteristic: it is zero if equivalently

• its underlying $\mathbb{Z}$-module is a flat module,

• its underlying abelian group is a torsion-free group.

• The unique ring homomorphism from $\mathbb{Z}$ to $R$ is an injection.

If a mathematical construct involves a “base ring”, e.g. an algebraic variety, then we say that it is in characteristic zero, if its base ring is.

Examples

The basic example of a ring of characteristic zero is the field $\mathbb{Q}$ of rational numbers. Therefore one could be tempted to define a ring (or even super ring) $R\supset \mathbb{Q}$ of characteristic $0$ as one containing the rationals. A ring of this form is exactly a $\mathbb{Q}$-algebra. While every $\mathbb{Q}$-algebra is a ring of characteristic $0$, some rings of characteristic $0$ are not $\mathbb{Q}$-algebras, for instance the ring $\mathbb{Z}$.

The basic example of an algebraically closed field of characteristic zero is the field $\mathbb{C}$ of complex numbers.

In model theory, there is a first-order theory of fields: every (commutative) field is a model. There is a transfer principle called Lefschetz principle which says: every sentence expressed in the first order theory of fields which is true for complex numbers is true for every algebraically closed field of characteristic zero. It is named after Solomon Lefschetz who used it in algebraic geometry, reasoning topologically for other algebarically closed fields of characteristic zero. The formalization and its proof are due Alfred Tarski.

Related concepts

• positive characteristic