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Center (algebra)
The term
center
or
centre
is used in various
contexts
in
abstract algebra
to
denote
the set of all those
elements
that
commute
with all other elements.
The
center of a group
G
consists of all those elements
x
in
G
such that
xg
=
gx
for all
g
in
G
. This is a
normal subgroup
of
G
.
The similarly named
notion
for a
semigroup
is defined
likewise
and it is a
subsemigroup
.
The
center of a ring
R
is the
subset
of
R
consisting of
all those elements
x
of
R
such that
xr
=
rx
for all
r
in
R
. The center is a
commutative
subring
of
R
.
The center of a
Lie algebra
L
consists of all those elements
x
in
L
such that = 0 for all
a
in
L
. This is an
ideal
of the
Lie
algebra
L
.
