Editing Glossary of ring theory (section)

== I ==
{{glossary}}

{{term|1=ideal}}
{{defn|A [[left ideal]] ''I'' of ''R'' is an additive subgroup of ''R'' such that {{nowrap|''aI'' ⊆ ''I''}} for all {{nowrap|''a'' ∈ ''R''}}. A ''right ideal'' is a subgroup of ''R'' such that {{nowrap|''Ia'' ⊆ ''I''}} for all {{nowrap|''a'' ∈ ''R''}}. An ''ideal'' (sometimes called a ''two-sided ideal'' for emphasis) is a subgroup that is both a left ideal and a right ideal.}}

{{term|1=idempotent}}
{{defn|1=An element ''r'' of a ring is [[Idempotent element (ring theory)|idempotent]] if {{nowrap|1=''r''{{i sup|2}} = ''r''}}.}}

{{term|1=integral domain}}
{{defn|1="'''[[integral domain]]'''" or "'''entire ring'''" is another name for a [[commutative domain]]; i.e., a nonzero [[commutative ring]] with no [[zero divisor]]s except 0.}}

{{term|1=invariant}}
{{defn|1=A ring ''R'' has [[invariant basis number]] if ''R''<sup>''m''</sup> isomorphic to ''R''<sup>''n''</sup> as [[module (mathematics)|''R''-modules]] implies {{nowrap|1=''m'' = ''n''}}.}}

{{term|1=irreducible}}
{{defn|1=An element ''x'' of an integral domain is [[Irreducible element|irreducible]] if it is not a unit and for any elements ''a'' and ''b'' such that {{nowrap|1=''x'' = ''ab''}}, either ''a'' or ''b'' is a unit. Note that every prime element is irreducible, but not necessarily vice versa.}}

{{glossary end}}