Editing Glossary of ring theory (section)

== C ==
{{glossary}}
{{term|1=category}}
{{defn|1=The [[category of rings]] is a category where the objects are (all) the rings and where the morphisms are (all) the ring homomorphisms.}}

{{term|1=centre}}
{{defn|no=1|1=An element ''r'' of a ring ''R'' is ''central'' if {{nowrap|1=''xr'' = ''rx''}} for all ''x'' in ''R''. The set of all central elements forms a [[subring]] of ''R'', known as the [[center (ring theory)|centre]] of ''R''.}}
{{defn|no=2|1=A [[central algebra]] is an associative algebra over the centre.}}
{{defn|no=3|1=A [[central simple algebra]] is a central algebra that is also a simple ring.}}

{{term|1=centralizer}}
{{defn|no=1|The [[centralizer (ring theory)|centralizer]] of a subset ''S'' of a ring is the subring of the ring consisting of the elements commuting with the elements of ''S''. For example, the centralizer of the ring itself is the centre of the ring.}}
{{defn|no=2|The [[double centralizer]] of a set is the centralizer of the centralizer of the set. Cf. [[double centralizer theorem]].}}

{{term|1=characteristic}}
{{defn|no=1|1=The [[Characteristic (algebra)|characteristic]] of a ring is the smallest positive integer ''n'' satisfying ''nx'' = 0 for all elements ''x'' of the ring, if such an ''n'' exists. Otherwise, the characteristic is 0.}}
{{defn|no=2|1=The [[characteristic subring]] of ''R'' is the smallest subring (i.e., the unique minimal subring). It is necessary the image of the unique ring homomorphism {{nowrap|'''Z''' → ''R''}} and thus is isomorphic to '''Z'''/''n'' where ''n'' is the characteristic of ''R''.}}

{{term|1=change}}
{{defn|A [[change of rings]] is a functor (between appropriate categories) induced by a ring homomorphism.}}

{{term|1=Clifford algebra}}
{{defn|1=A [[Clifford algebra]] is a certain associative algebra that is useful in geometry and physics.}}

{{term|coherent}}
{{defn|1=A left [[coherent ring]] is a ring such that every finitely generated left ideal of it is a finitely presented module; in other words, it is [[coherent module|coherent]] as a left module over itself.}}

{{term|1=commutative}}
{{defn|no=1|1=A ring ''R'' is [[commutative ring|commutative]] if the multiplication is commutative, i.e. {{nowrap|1=''rs'' = ''sr''}} for all {{nowrap|''r'',''s'' ∈ ''R''}}.}}
{{defn|no=2|1=A ring ''R'' is [[skew-commutative ring]] if {{nowrap|1=''xy'' = (−1)<sup>''ε''(''x'')''ε''(''y'')</sup>''yx''}}, where ''ε''(''x'') denotes the parity of an element ''x''.}}
{{defn|no=3|1=A commutative algebra is an associative algebra that is a commutative ring.}}
{{defn|no=4|1=[[Commutative algebra]] is the theory of commutative rings.}}

{{glossary end}}