Last Updated：2025/12/30
Sentence
The
unique
maximal
ideal
of
a
(commutative)
local
ring
contains
all
of
the
zero
divisors
of
such
ring,
and
all
elements
of
the
ring
outside
of
it
are
units.
Then
in
a
local
ring,
the
sum
of
any
two
zero
divisors
is
also
a
zero
divisor.
Contrapositively,
if
two
ring
elements
add
up
to
a
unit
then
one
of
them
must
be
a
unit
as
well.
A
simple
example
of
a
local
ring
is
ℤ₈.
Quizzes for review
The unique maximal ideal of a (commutative) local ring contains all of the zero divisors of such ring, and all elements of the ring outside of it are units. Then in a local ring, the sum of any two zero divisors is also a zero divisor. Contrapositively, if two ring elements add up to a unit then one of them must be a unit as well. A simple example of a local ring is ℤ₈.
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