{{Short description|Abstract algebra concept}} In abstract algebra, a nonzero 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 domains and simple rings.
Although this article discusses the above definition, '''prime ring''' may also refer to the minimal non-zero subring of a field, which is generated by its identity element 1, and determined by its characteristic. For a characteristic 0 field, the prime ring is the 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>
==Equivalent definitions== A ring ''R'' is prime if and only if the zero ideal {0} is a prime ideal in the noncommutative sense.
This being the case, the equivalent conditions for prime ideals yield the following equivalent conditions for ''R'' to be a prime ring: *For any two ideals ''A'' and ''B'' of ''R'', ''AB'' = {0} implies ''A'' = {0} or ''B'' = {0}. *For any two ''right'' ideals ''A'' and ''B'' of ''R'', ''AB'' = {0} implies ''A'' = {0} or ''B'' = {0}. *For any two ''left'' ideals ''A'' and ''B'' of ''R'', ''AB'' = {0} implies ''A'' = {0} or ''B'' = {0}.
Using these conditions it can be checked that the following are equivalent to ''R'' being a prime ring: *All nonzero right ideals are faithful as right ''R''-modules. *All nonzero left ideals are faithful as left ''R''-modules.
== Examples ==
* Any domain is a prime ring. * Any simple ring is a prime ring, and more generally: every left or right primitive ring is a prime ring. * Any matrix ring over an integral domain is a prime ring. In particular, the ring of 2 × 2 integer matrices is a prime ring. <!-- Apologies about deleting the prime PRIR example as incorrect: it might be correct. It'smm still not appropriate for the examples section. -->
== Properties ==
* A commutative ring is a prime ring if and only if it is an integral domain. * A nonzero ring is prime if and only if the monoid of its ideals lacks zero divisors. * The ring of matrices over a prime ring is again a prime ring.
==Notes== <references/>
==References== *{{Citation | last1=Lam | first1=Tsit-Yuen |author-link=Tsit Yuen Lam | title=A First Course in Noncommutative Rings | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | isbn=978-0-387-95325-0 |mr=1838439 | year=2001}}
{{DEFAULTSORT:Prime Ring}} Category:Ring theory