modules and rings by john dauns

By john dauns

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In fact e ∈ / m for every e ∈ α . (d) The correspondence in (c) sets up a bijection between the sets P(A) and Max(A) . (e) For every point α of A , there is a unique simple A-module V (up to isomorphism) such that e · V = 0 for some e ∈ α . In fact e · V = 0 for every e ∈ α and V ∼ = Ae . (f) The correspondence in (e) sets up a bijection between the sets P(A) and Irr(A) . (g) Any two primitive decompositions of 1A are conjugate under A∗ . The theorem on lifting idempotents allows us to generalize (c)–(g) to any finite dimensional k-algebra, but we shall consider in Section 3 an even more general situation.

In other words mα is the size of the matrix algebra S(α) , that is, dimk (S(α)) = m2α . We record these facts for later use. 15) PROPOSITION. Let A be an O-algebra and let mα be the multiplicity of a point α ∈ P(A) . (a) mα = dimk (V (α)) , where V (α) is a simple A-module corresponding to α . (b) m2α = dimk (S(α)) , where S(α) is the simple quotient of A corresponding to α . § 4 . Idempotents and points 29 For the reasons above, the simple quotient S(α) corresponding to a point α is called the multiplicity algebra of the point α .

1. The details are left as an exercise for the reader. (e) This is a direct consequence of (b), (c) and (d). (f) This is an easy exercise. 11 which is proved in the next section. 1) Let M be a left ideal in an O-algebra A . Prove that either M contains an idempotent or we have M ⊆ J(A) . 2) Let e and f be two idempotents of an O-algebra A . Prove that if e = ab and f = ba for some a, b ∈ A , then e and f are conjugate (and conversely). 3) Let a and b be two elements of an O-algebra A such that ab = 1 .

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