Ring with operators

ring with domain of operators $\Sigma$

A ring on which a an action ( "multiplication" ) of elements of the ring by elements from a fixed set $\Sigma$ is defined (an external law of composition), such that the following axioms are satisfied:

$$\tag{1 } ( a + b) \alpha = a \alpha + b \alpha ,$$

$$\tag{2 } ( ab) \alpha = ( a \alpha ) b = a( b \alpha ),$$

where $\alpha$ is an element of $\Sigma$ while $a$, $b$, $a \alpha$, $b \alpha$ 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 $\Sigma$, or, more succinctly, a $\Sigma$- operator ring, can also be treated as a universal algebra with two binary operations (addition and multiplication) and with a set $\Sigma$ of unary operations linked by the usual ring identities as well as by the identities (1) and (2). The concepts of a $\Sigma$- permissible subring, a $\Sigma$- permissible ideal, a $\Sigma$- operator isomorphism, and a $\Sigma$- operator homomorphism can be defined in the same way as for groups with operators (cf. Operator group). If a $\Sigma$- operator ring $R$ possesses a unit element, then all ideals and all one-sided ideals of the ring $R$ are $\Sigma$- permissible.

A ring $R$ is called a ring with a ring of operators $\Sigma$ if it is a $\Sigma$- operator ring whose domain of operators $\Sigma$ is itself an associative ring, while for any $\alpha , \beta \in \Sigma$ and $a \in R$ the following equalities hold:

$$\tag{3 } a( \alpha + \beta ) = a \alpha + a \beta ,$$

$$\tag{4 } a( \alpha \beta ) = ( a \alpha ) \beta .$$

A ring with a ring of operators can also be defined as a ring which is simultaneously a $\Sigma$- module and which satisfies axiom (2). Every ring can naturally be considered as an operator ring over the ring of integers.

For all $a$ from $R$ and $\alpha , \beta$ from $\Sigma$, the element $a( \alpha \beta - \beta \alpha )$ is an annihilator of $R$. Therefore, if $R$ is a ring with operators without annihilators, then its ring of operators $\Sigma$ 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)

Thus, the bilinearity properties (1), (2) and the module properties (3), (4) are practically incompatible for rings $A$ with a non-commutative ring of operators $R$ in that $b \cdot a ( \alpha \beta - \beta \alpha ) = 0$ for all $a, b \in A$, $\alpha , \beta \in R$. 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 $( ab) \alpha = a( b \alpha )$. Cf. also Algebra and Ring.