Editing Prime ring (section)

{{Short description|Abstract algebra concept}}
In [[abstract algebra]], a [[zero ring|nonzero]] [[ring (mathematics)|ring]] ''R'' is a '''prime ring''' if for any two elements ''a'' and ''b'' of ''R'', ''arb'' = 0 for all ''r'' in ''R'' implies that either ''a'' = 0 or ''b'' = 0. This definition can be regarded as a simultaneous generalization of both [[integral domain]]s and [[simple ring]]s.

Although this article discusses the above definition, '''prime ring''' may also refer to the minimal non-zero [[subring]] of a [[field (mathematics)|field]], which is generated by its identity element 1, and determined by its [[characteristic (algebra)|characteristic]]. For a characteristic 0 field, the prime ring is the [[Integer#Algebraic_properties|integers]], and for a characteristic ''p'' field (with ''p'' a [[prime number]]) the prime ring is the [[finite field]] of order ''p'' (cf. [[Prime field]]).<ref name="lang">Page 90 of {{Lang Algebra|edition=3}}</ref>