• The zero ring is the unique ring in which the
additive identity 0 and
multiplicative identity 1 coincide. (Proof: If in a ring
R, then for all
r in
R, we have . The proof of the last equality is found
here.) • The zero ring is commutative. • The element 0 in the zero ring is a
unit, serving as its own
multiplicative inverse. • The
unit group of the zero ring is the
trivial group {0}. • 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. 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. • For each ring
A, there is a unique
ring homomorphism from
A to the zero ring. Thus the zero ring is a
terminal object in the
category of rings. • 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. • The zero ring is the unique ring of
characteristic 1. • The only
module for the zero ring is the zero module. It is free of rank א for any
cardinal number א. • The zero ring is not a
local ring. It is, however, a
semilocal ring. • The zero ring is
Artinian and (therefore)
Noetherian. • The
spectrum of the zero ring is the empty
scheme. • The
Krull dimension of the zero ring is −∞. • The zero ring is
semisimple but not
simple. • The zero ring is not a
central simple algebra over any field. • The
total quotient ring of the zero ring is itself. == Constructions ==