Zero ring

In abstract algebra, a zero ring, also called a trivial ring, is a ring containing only one element. This element is simultaneously the additive identity (zero) and the multiplicative identity (one). Consequently, in a zero ring, 0 = 1.

More formally, a set {0} equipped with the binary operations + and ⋅ such that 0 + 0 = 0 and 0 ⋅ 0 = 0 forms a zero ring.

The zero ring is the terminal object in the category of rings. This means that for any ring R, there exists a unique ring homomorphism from R to the zero ring, mapping every element of R to 0.

Because the additive identity and multiplicative identity are the same, a zero ring does not satisfy the axiom that 0 ≠ 1 which is often included in the definition of a ring with identity. However, depending on the specific definition used, the zero ring may or may not be considered a ring with identity. If the requirement 0 ≠ 1 is omitted, then the zero ring is a ring with identity.

The zero ring is a commutative ring and a field if and only if 0 ≠ 1 is not a requirement in the definition of a field.

The zero ring is useful as an object satisfying certain universal properties in category theory, particularly in the context of ring theory.