Science Fair Project Encyclopedia
Annihilator (ring theory)
Annihilators are a concept that occurs in ring theory, a branch of mathematics. Let R be a ring, and let M be a left R-module. Choose a subset S of M. The annihilator AnnRS of S is the set of all elements r in R such that for each s in S, rs=0.
The annihilator of a single element x is usually written AnnRx instead of AnnR{x}. If the ring R can be understood from the context, the subscript R is usually omitted.
Annihilators are always one-sided ideals of their ring: If a and b both annihilate S, then for each s in S, (a+b)s=as+bs=0, and for any c in R, (ca)s=c(as)=c0=0. The annihilator of M is even a two-sided ideal: (ac)s=a(cs)=0, since cs is another element of M.
M is always a faithful R/AnnRM-module.
Last updated: 05-21-2005 10:36:35
10-26-2009 08:16:03
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details