Editing Glossary of ring theory (section)

== D ==
{{glossary}}

{{term|1=derivation}}
{{defn|no=1|1=A [[derivation of an algebra|derivation]] of a possibly-non-associative algebra ''A'' over a commutative ring ''R'' is an ''R''-linear endomorphism that satisfies the [[Product rule|Leibniz rule]].}}
{{defn|no=2|1=The [[derivation algebra]] of an algebra ''A'' is the subalgebra of the endomorphism algebra of ''A'' that consists of derivations.}}

{{term|1=differential}}
{{defn|1=A [[differential algebra]] is an algebra together with a derivation.}}

{{term|1=direct}}
{{defn|1=A [[direct product ring|direct product]] of a family of rings is a ring given by taking the [[cartesian product]] of the given rings and defining the algebraic operations component-wise.}}

{{term|1=divisor}}
{{defn|no=1|1=In an [[integral domain]] ''R'',{{clarify|Do we need to assume ''R'' is an integral domain?|date=January 2020}} an element ''a'' is called a [[Divisibility (ring theory)|divisor]] of the element ''b'' (and we say ''a'' ''divides'' ''b'') if there exists an element ''x'' in ''R'' with {{nowrap|1=''ax'' = ''b''}}.}}
{{defn|no=2|1=An element ''r'' of ''R'' is a ''left [[zero divisor]]'' if there exists a nonzero element ''x'' in ''R'' such that {{nowrap|1=''rx'' = 0}} and a ''right zero divisor'' or if there exists a nonzero element ''y'' in ''R'' such that {{nowrap|1=''yr'' = 0}}. An element ''r'' of ''R'' is a called a ''two-sided zero divisor'' if it is both a left zero divisor and a right zero divisor.}}

{{term|1=division}}
{{defn|1=A [[division ring]] or skew field is a ring in which every nonzero element is a unit and {{nowrap|1 ≠ 0}}.}}

{{term|1=domain}}
{{defn|1=A [[Domain (ring theory)|domain]] is a nonzero ring with no zero divisors except 0. For a historical reason, a commutative domain is called an [[integral domain]].}}

{{glossary end}}