Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Prime ring
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
{{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> ==Equivalent definitions== A ring ''R'' is prime [[if and only if]] the [[zero ideal]] {0} is a [[Prime_ideal#Prime_ideals_for_noncommutative_rings|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 [[ideal (ring theory)|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 module|faithful]] as right ''R''-modules. *All nonzero left ideals are faithful as left ''R''-modules. == Examples == * Any [[domain (ring theory)|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 Γ 2 integer [[matrix (mathematics)|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 [[ideal (ring theory)|ideal]]s lacks [[zero divisor]]s. * The ring of matrices over a prime ring is again a prime ring. ==Notes== <references/> ==References== *{{Citation | last1=Lam | first1=Tsit-Yuen | 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]]
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)
Pages transcluded onto the current version of this page
(
help
)
:
Template:Citation
(
edit
)
Template:Lang Algebra
(
edit
)
Template:Short description
(
edit
)