A
ring
on which a an action
( "multiplication" )
of elements of
the ring by elements from a fixed set
is defined (an external law of composition),
such that the following axioms are satisfied:
where
is an element of
while
,
,
,
are elements of the ring. In this way, the operators
act as endomorphisms of the additive group, commuting with multiplication by
an element of the ring. A ring with domain of operators
,
or, more succinctly, a
-operator ring,
can also be treated as a
universal algebra
with two binary operations (addition and multiplication) and with a set
of unary operations linked by the usual ring identities as
well as by the identities
(1)
and
(2).
The concepts of a
-permissible subring,
a
-permissible ideal,
a
-operator isomorphism,
and a
-operator homomorphism
can be defined in the same way as for groups with operators (cf.
Operator group).
If a
-operator
ring
possesses a unit element, then all ideals and all one-sided ideals of the ring
are
-permissible.
A ring
is called a
ring with a ring of operators
if it is a
-operator
ring whose domain of operators
is itself an associative ring, while for any
and
the following equalities hold:
A ring with a ring of operators can also be defined as a ring which is simultaneously a
-module
and which satisfies axiom
(2).
Every ring can naturally be
considered as an operator ring over the ring of integers.
For all
from
and
from
,
the element
is an
annihilator
of
.
Therefore, if
is a ring with operators without annihilators, then its ring of operators
must be commutative.
The most commonly studied rings with operators are those with
an associative-commutative ring of operators possessing a unit element.
This type of ring is usually called an
algebra over a commutative ring,
and also a
linear algebra.
The most commonly studied linear algebras are those over fields;
the theory of these algebras is evolving in parallel
with the general theory of rings (without operators).