View source for Ring with operators

''ring with domain of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r0824501.png" />''

A [[Ring|ring]] on which a an action ( "multiplication" ) of elements of the ring by elements from a fixed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r0824502.png" /> is defined (an external law of composition), such that the following axioms are satisfied:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r0824503.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r0824504.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r0824505.png" /> is an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r0824506.png" /> while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r0824507.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r0824508.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r0824509.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245010.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245011.png" />, or, more succinctly, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245013.png" />-operator ring, can also be treated as a [[Universal algebra|universal algebra]] with two binary operations (addition and multiplication) and with a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245014.png" /> of unary operations linked by the usual ring identities as well as by the identities (1) and (2). The concepts of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245016.png" />-permissible subring, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245018.png" />-permissible ideal, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245020.png" />-operator isomorphism, and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245022.png" />-operator homomorphism can be defined in the same way as for groups with operators (cf. [[Operator group|Operator group]]). If a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245023.png" />-operator ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245024.png" /> possesses a unit element, then all ideals and all one-sided ideals of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245025.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245026.png" />-permissible.

A ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245027.png" /> is called a ring with a ring of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245028.png" /> if it is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245029.png" />-operator ring whose domain of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245030.png" /> is itself an associative ring, while for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245032.png" /> the following equalities hold:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

A ring with a ring of operators can also be defined as a ring which is simultaneously a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245035.png" />-module and which satisfies axiom (2). Every ring can naturally be considered as an operator ring over the ring of integers.

For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245036.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245038.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245039.png" />, the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245040.png" /> is an [[Annihilator|annihilator]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245041.png" />. Therefore, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245042.png" /> is a ring with operators without annihilators, then its ring of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245043.png" /> 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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,   "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR></table>



====Comments====
Thus, the bilinearity properties (1), (2) and the module properties (3), (4) are practically incompatible for rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245044.png" /> with a non-commutative ring of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245045.png" /> in that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245046.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245048.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082450/r08245049.png" />. Cf. also [[Algebra|Algebra]] and [[Ring|Ring]].