Zero ring

The zero ring or trivial ring is in the mathematics of up to isomorphism uniquely defined ring , the only of the zero element exists. The zero element is thus also the one element of the ring. The zero ring has a number of special properties, for example it is the only ring in which every element is a unit and the only ring with one in which there is no maximum ideal . In the category of rings with one, the zero ring is the terminal object and in the category of all rings it is the zero object .

definition

The zero ring is a ring consisting of the one-element set provided with the only possible addition given by ${\ displaystyle (\ {0 \}, +, \ cdot)}$ ${\ displaystyle \ {0 \}}$

{\ displaystyle 0 + 0 = 0}

and the only possible multiplication given by

{\ displaystyle 0 \ cdot 0 = 0}

.

The element is therefore at the same time the zero element and the one element of the ring. ${\ displaystyle 0}$

properties

The zero ring is a commutative ring with one . Since the zero element is not a zero divisor , the zero ring has no zero divisors . The null ring is the only ring in which the null element is a unit , and even the only ring in which every element is a unit. According to Zorn's lemma , it is the only unitary ring in which there is no maximal ideal .

Every ring in which holds is isomorphic to the zero ring, because then holds ${\ displaystyle R}$ ${\ displaystyle 1 = 0}$

{\ displaystyle r = 1 \ cdot r = 0 \ cdot r = 0}

for all elements . One encounters the zero ring, for example, when one factorizes a ring according to itself , or by localizing it according to a multiplicative system that contains the zero element . ${\ displaystyle r \ in R}$

The zero ring is the only ring in which division (the inverse of multiplication) is completely unrestricted for all elements, and d. H. in this case also by 0, is possible: The result is 0.

The zero ring is not a body , as these structures are always required. It is also not an integrity ring , since it is isomorphic to any ring , but the whole ring is not a prime ideal . ${\ displaystyle 1 \ neq 0}$ ${\ displaystyle R}$ ${\ displaystyle R / R}$

Category theory

In the category of rings with one, the zero ring is the terminal object , i.e. there is exactly one morphism in the zero ring from each ring. Furthermore, every morphism out of the zero ring is already an isomorphism .

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

literature

Michael Artin : Algebra . Birkhäuser, Basel et al. 1998, ISBN 3-7643-5938-2 .

Individual evidence

↑ ^a ^b Artin: Algebra . 1998, p. 396 .

Web links

Margherita Barile: Trivial Ring . In: MathWorld (English).

[artin396-1] Artin: Algebra . 1998, p. 396 .