Editing Glossary of ring theory (section)

== N ==
{{glossary}}

{{term|1=nearring}}
{{defn|1=A '''[[nearring]]''' is a structure that is a group under addition, a [[semigroup]] under multiplication, and whose multiplication distributes on the right over addition.}}

{{term|1=nil}}
{{defn|no=1|1=A [[nil ideal]] is an ideal consisting of nilpotent elements.}}
{{defn|no=2|1=The (Baer) [[nilradical of a ring|upper nil radical]] is the sum of all nil ideals.}}
{{defn|no=3|1=The (Baer) [[nilradical of a ring|lower nil radical]] is the intersection of all prime ideals. For a commutative ring, the upper nil radical and the lower nil radical coincide.}}

{{term|1=nilpotent}}
{{defn|no=1|An element ''r'' of ''R'' is [[nilpotent element|nilpotent]] if there exists a positive integer ''n'' such that {{nowrap|1=''r''{{i sup|''n''}} = 0}}.}}
{{defn|no=2|1=A [[nil ideal]] is an ideal whose elements are nilpotent elements.}}
{{defn|no=3|1=A [[nilpotent ideal]] is an ideal whose [[product of ideals|power]] ''I''<sup>''k''</sup> is {0} for some positive integer ''k''. Every nilpotent ideal is nil, but the converse is not true in general.}}
{{defn|no=4|1=The [[Nilradical of a ring|nilradical]] of a commutative ring is the ideal that consists of all nilpotent elements of the ring. It is equal to the intersection of all the ring's [[prime ideal]]s and is contained in, but in general not equal to, the ring's Jacobson radical.}}

{{term|Noetherian}}
{{defn|1=A left [[Noetherian ring]] is a ring satisfying the [[ascending chain condition]] for left ideals. A ''right Noetherian'' is defined similarly and a ring that is both left and right Noetherian is ''Noetherian''. A ring is left Noetherian if and only if all its left ideals are finitely generated; analogously for right Noetherian rings.}}

{{term|1=null}}
{{defn|1='''null ring''': See {{gli|rng of square zero}}.}}

{{glossary end}}