By M. Anderson

ISBN-10: 1584885157

ISBN-13: 9781584885153

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Extra resources for A first course in abstract algebra: rings, groups, and fields

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Now let w be an Ω -word with (w) ≥ 2, and let k ≥ 1 be minimal with w ∈ Xk . 2 w can be written uniquely as w = ω (w1 , . . , wn ) where n ≥ 1, ω ∈ Ω (n), and wi ∈ Xk−1 . Assuming that f¯(wi ) is defined for 1 ≤ i ≤ n, let f¯(w) = ω ( f¯(w1 ), . . , f¯(wn )). This defines f¯ on FΩ (X), and clearly f¯ is a morphism of Ω -algebras. Suppose that w = (y1 , . . , ym ) is an Ω -word where yi ∈ Ω ∪ X. If {y1 , . . , ym } ∩ X ⊆ {x1 , . . , xn } we will write w = w(x1 , . . , xn ) to indicate that x1 , .

B) Show that each Ω -algebra A is a subdirect product of a family of subdirectly irreducible Ω -algebras. (If a = b in A let ∼ be a congruence on A that is maximal with respect to (a, b) ∈ ∼). 13. Show that each variety in A (Ω ) is hereditary, homomorphically closed and productive. 14. Show that each free C -algebra F is projective; that is, if A, B ∈ C , where C is a class of Ω -algebras, and h : A −→ B is an epimorphism, then for each morphism f : F −→ B there is a morphism g : F −→ A with hg = f : 15.

An Ω -algebra is a set A together with a sequence of functions n Ω (n) −→ A(A ) . When no confusion is likely an Ω -algebra A will be referred to as just an algebra. Each ω ∈ Ω (n) determines an n-ary operation on A, and the value of this operation at (a1 , . . , an ) ∈ An will be denoted by ω (a1 , . . , an ). If ω is a 0-ary operator, then by / and ω determines some distinguished element or constant definition A0 = A0/ = {0}, ωA of A. For this reason a 0-ary operator is frequently called a constant operator.

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A first course in abstract algebra: rings, groups, and fields by M. Anderson


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