nLab trivial ring

Definition

The trivial ring, or zero ring, is the ring with a single element, which is both $0$ and $1$.

One usually denote the trivial ring as $0$ or $\{0\}$, even though $1$ or $\{1\}$ would make as much sense.

The trivial rig or zero rig is the rig with a single element, which is both $0$ and $1$. By definition of neutral element, this single element is its own additive inverse, making the trivial rig the same as the trivial ring.

Properties

• The trivial ring is the only ring in which $0 = 1$, since if that is the case then it follows for all elements $x$ that $x = 1 x = 0 x = 0$.

• The trivial ring is the terminal object in Rings. It is both terminal and initial (hence a zero object) in the category of nonunital rings, but it is not initial in Rings (defined as the category of unital rings and unital ring homomorphisms). In fact, there are no unital ring homomorphisms from the trivial ring to any nontrivial ring, hence it is a strict terminal object there.

• The trivial ring is an example of a trivial algebra.

• Every ring is a 0-truncated ring groupoid, and the trivial ring is the terminal ring groupoid.

• If one takes the definition of a field from LombardiQuitté2010, then the trivial ring is a terminal object in the category Field of fields.

• The trivial ring is a discrete ring and is the terminal object of both the category of tight apartness rings and the category of discrete rings.

See also

• zero object

References

• Henri Lombardi, Claude Quitté (2010): Commutative algebra: Constructive methods (Finite projective modules) Translated by Tania K. Roblo, Springer (2015) (doi:10.1007/978-94-017-9944-7, pdf)