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.