In
ring theory, a branch of
mathematics, the zero ring[1][2][3][4][5] or trivial ring is the unique
ring (up to
isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any
rng of square zero, i.e., a
rng in which xy = 0 for all x and y. This article refers to the one-element ring.)
The zero ring, denoted {0} or simply 0, consists of the
one-element set {0} with the
operations + and · defined such that 0 + 0 = 0 and 0 · 0 = 0.
Properties
The zero ring is the unique ring in which the
additive identity 0 and
multiplicative identity 1 coincide.[1][6] (Proof: If 1 = 0 in a ring R, then for all r in R, we have r = 1r = 0r = 0. The proof of the last equality is found here.)
The element 0 in the zero ring is not a
zero divisor.
The only
ideal in the zero ring is the zero ideal {0}, which is also the unit ideal, equal to the whole ring. This ideal is neither
maximal nor
prime.
The zero ring is generally excluded from
fields, while occasionally called as the trivial field. Excluding it agrees with the fact that its zero ideal is not maximal. (When mathematicians speak of the "
field with one element", they are referring to a non-existent object, and their intention is to define the category that would be the category of schemes over this object if it existed.)
The zero ring is generally excluded from
integral domains.[7] Whether the zero ring is considered to be a
domain at all is a matter of convention, but there are two advantages to considering it not to be a domain. First, this agrees with the definition that a domain is a ring in which 0 is the only zero divisor (in particular, 0 is required to be a zero divisor, which fails in the zero ring). Second, this way, for a positive integer n, the ring Z/nZ is a domain if and only if n is prime, but 1 is not prime.
If A is a nonzero ring, then there is no ring homomorphism from the zero ring to A. In particular, the zero ring is not a
subring of any nonzero ring.[8]
In
ring theory, a branch of
mathematics, the zero ring[1][2][3][4][5] or trivial ring is the unique
ring (up to
isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any
rng of square zero, i.e., a
rng in which xy = 0 for all x and y. This article refers to the one-element ring.)
The zero ring, denoted {0} or simply 0, consists of the
one-element set {0} with the
operations + and · defined such that 0 + 0 = 0 and 0 · 0 = 0.
Properties
The zero ring is the unique ring in which the
additive identity 0 and
multiplicative identity 1 coincide.[1][6] (Proof: If 1 = 0 in a ring R, then for all r in R, we have r = 1r = 0r = 0. The proof of the last equality is found here.)
The element 0 in the zero ring is not a
zero divisor.
The only
ideal in the zero ring is the zero ideal {0}, which is also the unit ideal, equal to the whole ring. This ideal is neither
maximal nor
prime.
The zero ring is generally excluded from
fields, while occasionally called as the trivial field. Excluding it agrees with the fact that its zero ideal is not maximal. (When mathematicians speak of the "
field with one element", they are referring to a non-existent object, and their intention is to define the category that would be the category of schemes over this object if it existed.)
The zero ring is generally excluded from
integral domains.[7] Whether the zero ring is considered to be a
domain at all is a matter of convention, but there are two advantages to considering it not to be a domain. First, this agrees with the definition that a domain is a ring in which 0 is the only zero divisor (in particular, 0 is required to be a zero divisor, which fails in the zero ring). Second, this way, for a positive integer n, the ring Z/nZ is a domain if and only if n is prime, but 1 is not prime.
If A is a nonzero ring, then there is no ring homomorphism from the zero ring to A. In particular, the zero ring is not a
subring of any nonzero ring.[8]