Annihilator (ring theory)

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In mathematics, specifically module theory, annihilators are a concept that generalizes torsion and orthogonal complement.

Contents 1 Definition 2 Properties 3 Relations to other properties of rings 4 References 5 See also

[edit] Definition

Let R be a ring, and let M be a left R-module. Choose a subset S of M. The annihilator, Ann_RS, of S is the set of all elements r in R such that for each s in S, rs = 0: it is the set of all elements that annihilate S (the elements for which S is torsion).

More generally, given a bilinear map of modules , the annihilator of a subset is the set of all elements in N that annihilate S:

Conversely, given , one can define an annihilator as a subset of M.

The annihilator gives a Galois connection between subsets of M and N, and the associated closure operator is stronger than the span. In particular:

• annihilators are submodules
• 
• 

An important special case is in the presence of a nondegenerate form on a vector space, particularly an inner product: then the annihilator associated to the map is called the orthogonal complement.

[edit] Properties

The annihilator of a single element x is usually written Ann_Rx instead of Ann_R{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/Ann_RM-module.

[edit] Relations to other properties of rings

• The set of (left) zero divisors D_S of S can be written as

In particular D is the set of (left) zero divisors of R when S = R and R acts on itself as a left R-module.

[edit] References

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[edit] See also

• socle