Trivial ring

The zero ring or trivial ring is in the mathematics of up to isomorphism uniquely determined ring, which consists only of the zero element. The zero element is at the same time also the identity element of the ring. The zero ring has a number of special properties, it is for example the only ring in which every element is a unit, and the only ring with one in which there is no maximal ideal. In the category of rings with unity, the zero ring terminal object in the category of all rings and the null object.

Definition

The zero ring is a ring consisting of the singleton provided with the only possible addition given by

And the only possible multiplication given by

The element is thus simultaneously the zero element and the unit element of the ring.

Properties

The zero ring is a commutative ring with unity. Since the zero element is not a zero divisor, the zero ring is zero divisors. Zero is the only annular ring in which the zero element is a unit, and even single ring, in which each element is a unit. By the lemma of Zorn he is the only unitary ring in which there is no maximal ideal.

Each ring is considered in which is isomorphic to the zero ring, because then applies

For all elements. One encounters the zero ring, for example, if you factored a ring by himself, or by looking for a multiplicative system, which includes the zero element, localized.

The zero ring is not a body, there is always required for these structures. He is not integral domain, since it is isomorphic to for an arbitrary ring, but the whole ring is not a prime ideal.

Category theory

In the category of rings with unity, the zero ring terminal object, ie of each ring there is a unique morphism to the zero ring. Furthermore, each morphism is from the zero ring out already an isomorphism.

In the category of all rings of the zero ring is even the null object.

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