Editing Glossary of ring theory (section)

== P ==
{{glossary}}

{{term|1=perfect}}
{{defn|1=A ''left [[perfect ring]]'' is one satisfying the [[descending chain condition]] on ''right'' principal ideals. They are also characterized as rings whose flat left modules are all projective modules. Right perfect rings are defined analogously. Artinian rings are perfect.}}

{{term|1=polynomial}}
{{defn|no=1|1=A [[polynomial ring]] over a commutative ring ''R'' is a commutative ring consisting of all the polynomials in the specified variables with coefficients in&nbsp;''R''.}}
{{defn|no=2|A [[Polynomial ring#Differential and skew-polynomial rings|skew polynomial ring]]
: Given a ring ''R'' and an endomorphism {{nowrap|''σ'' ∈ End(''R'')}} of ''R''. The skew polynomial ring {{nowrap|''R''[''x''; ''σ'']}} is defined to be the set {{nowrap|{{mset|''a''<sub>''n''</sub>''x''<sup>''n''</sup> + ''a''<sub>''n''−1</sub>''x''<sup>''n''−1</sup> + ... + ''a''<sub>1</sub>''x'' + ''a''<sub>0</sub> {{!}} ''n'' ∈ '''N''', ''a''<sub>''n''</sub>, ''a''<sub>''n''−1</sub>, ..., ''a''<sub>1</sub>, ''a''<sub>0</sub> ∈ ''R''}}}}, with addition defined as usual, and multiplication defined by the relation {{nowrap|1=''xa'' = ''σ''(''a'')''x'' ∀''a'' ∈ ''R''}}.}}

{{term|1=prime}}
{{defn|no=1|1=An element ''x'' of an integral domain is a [[prime element]] if it is not zero and not a unit and whenever ''x'' divides a product ''ab'', ''x'' divides ''a'' or ''x'' divides ''b''.}}
{{defn|no=2|1=An ideal ''P'' in a [[commutative ring]] ''R'' is [[prime ideal|prime]] if {{nowrap|''P'' ≠ ''R''}} and if for all ''a'' and ''b'' in ''R'' with ''ab'' in ''P'', we have ''a'' in ''P'' or ''b'' in ''P''. Every maximal ideal in a commutative ring is prime.}}
{{defn|no=3|1=An ideal ''P'' in a (not necessarily commutative) ring ''R'' is prime if {{nowrap|''P'' ≠ ''R''}} and for all ideals ''A'' and ''B'' of ''R'', {{nowrap|''AB'' ⊆ ''P''}} implies {{nowrap|''A'' ⊆ ''P''}} or {{nowrap|''B'' ⊆ ''P''}}.  This extends the definition for commutative rings.}}
{{defn|no=4|'''[[prime ring]]''' : A [[zero ring|nonzero ring]] ''R'' is called a ''prime ring'' if for any two elements ''a'' and ''b'' of ''R'' with {{nowrap|1=''aRb'' = 0}}, we have either {{nowrap|1=''a'' = 0}} or {{nowrap|1=''b'' = 0}}. This is equivalent to saying that the zero ideal is a prime ideal (in the noncommutative sense.) Every [[simple ring]] and every [[domain (ring theory)|domain]] is a prime ring.}}

{{term|1=primitive}}
{{defn|no=1|1=A ''left [[primitive ring]]'' is a ring that has a [[faithful module|faithful]] [[simple module|simple]] [[module (mathematics)|left ''R''-module]]. Every [[simple ring]] is primitive.  Primitive rings are [[prime ring|prime]].}}
{{defn|no=2|1=An ideal ''I'' of a ring ''R'' is said to be [[primitive ideal|primitive]] if ''R''/''I'' is primitive.}}

{{term|1=principal}}
{{defn|1=A '''[[principal ideal]]''' : A ''principal left ideal'' in a ring ''R'' is a left ideal of the form ''Ra'' for some element ''a'' of ''R''. A ''principal right ideal'' is a right ideal of the form ''aR'' for some element ''a'' of ''R''. A ''principal ideal'' is a two-sided ideal of the form ''RaR'' for some element ''a'' of ''R''.}}

{{term|1=principal}}
{{defn|no=1|A [[principal ideal domain]] is an integral domain in which every ideal is principal.}}
{{defn|no=2|A [[principal ideal ring]] is a ring in which every ideal is principal.}}

{{glossary end}}