Let S be a nonempty subset of a vector space V over a division ring. The annihilator of S is the subset S⁰ of V* given by S° = {∫ ϵ V*| (s,∫) = 0 for all s ϵ S}.

(a) 0° = V*; v⁰ = O; S ≠ {0} =>S⁰ ≠ V*.

(b) If W is a subspace of V, then w0 is a subspace of V*.

(c) If W is a subspace of V and dim V is finite, then dim W 0 = dim V - dim W^o.

(d) Let W,V be as in (c). There is an isomorphism W* ≅ V* / W⁰•

(e) Let W,V be as in (c) and identify V with V** under the isomorphism 0 of Theorem 4.12. Then (W0) 0 = W c V**.

Data from Theorem 4.12

(i) For each a e A let 0(a): A*--+ R be the map defined by [O(a)J(f) = (a,J) e R. Statement (2) after Theorem 4.10 shows that 0(a) is a homomorphism of right R-modules (that is, 0(a) e A**). The map 0: A--+ A** given by a~ 0(a) is a left R-module homomorphism by (1) after Theorem 4.10. (ii) LetX be a basis of A. If a e A, then a = r1x1 + r2x2 + · · · + r.,_x,. (ri e R; Xi eX). If fJ(a) = 0, then for all fe A*, 

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0 = (a,f) = In particular, for f = fx; (j = 1,2,...,n), rixi,f Therefore, a = Σ rixi = 20x Σ Erring). 0 = Trixif;) = Σridij = 1;. Oxi = 0 and 0 is a monomorphism.