Ring with operators
ring with domain of operators
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:
![]() | (1) |
![]() | (2) |
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:
![]() | (3) |
![]() | (4) |
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).
References
[1] | A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) |
Comments
Thus, the bilinearity properties (1), (2) and the module properties (3), (4) are practically incompatible for rings with a non-commutative ring of operators
in that
for all
,
. This explains why algebras are usually only considered over commutative rings. Instead of algebra (over a ring) one also sometimes finds vector algebra. Both this phrase and the phrase linear algebra for an algebra over a ring are nowadays rarely used.
For algebras over non-commutative rings the bilinearity property (2) is weakened to . Cf. also Algebra and Ring.
Ring with operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ring_with_operators&oldid=17809