Prime ideal
From Encyclopedia of Mathematics
Revision as of 21:49, 11 December 2012 by Remag12 (talk | contribs) (→Comments: about m system ststement)
A two-sided ideal 
of a ring 
such that the inclusion 
for any two-sided ideals 
and 
of 
implies that either 
or 
. An ideal 
of a ring 
is prime if and only if the set 
is an 
-system, i.e. for any 
there exists an 
such that 
. An ideal 
of a ring 
is prime if and only if the quotient ring by it is a prime ring.
Comments
This assumes that the empty set is an m system by default.
References
| [a1] | L.H. Rowen, "Ring theory" , I , Acad. Press (1988) pp. 163 |
How to Cite This Entry:
Prime ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Prime_ideal&oldid=12225
Prime ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Prime_ideal&oldid=12225
This article was adapted from an original article by K.A. Zhevlakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article