Definition 1

A ring $K$ is called a field if it does not consist only of $0$ and every non-zero element of $K$ is invertible.

The set of non-zero elements of the field $K$, with multiplication, is a group, which is precisely the multiplicative group $K^*$ of the ring $K$ (§ 8, no. 1). The opposite ring of a field is a field. A field is called commutative if its multiplication is commutative; such a field is identified with its opposite. A non-commutative field is sometimes called a skew field.