By S. A. Amitsur, D. J. Saltman, George B. Seligman
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Additional resources for Algebraists' Homage: Papers in Ring Theory and Related Topics
Suppose a, b ∈ α ; β. Then there is a c ∈ G such that a α c β b. The claim is that a = ac−1 c ≡β ac−1 b ≡α cc−1 b = b. week 8 41 ac−1 ≡β ac−1 and c ≡β b imply (ac−1 )c ≡β (ac−1)b, and a ≡α c and c−1 b ≡α c−1 b imply a(c−1b) ≡α c(c−1b). So a, b ∈ β ; α. Hence α ; β ⊆ β ; α. The permutability of congruence relations on groups is a reflection of the fact that normal subuniverses permute under complex product, and hence that the relative product of two normal subuniverses is a normal subuniverse (recall the correspondence between congruences and normal subuniverses).
Mn a1 a2 a3 an−1 an Figure 16 To see this first let α = ΘM n (ai , aj ) with i = j. 1 = ai ∨ aj ≡α ai ∨ ai = ai = ai ∧ ai ≡α ai ∧ aj = 0. So ΘM n (ai, aj ) = ∇Mn . Now let α = ΘM n (ai, 0) and choose any j = i. 1 = ai ∨ aj ≡α 0 ∨ aj = aj . So ΘM n (aj , 1) ⊆ ΘM n (ai, 0) for all j = i. If i, j, k are all distinct, then aj ≡α 1 ≡α ak , and hence ∇Mn = ΘM n (aj , ak ) ⊆ ΘM n (ai , 0). Similarly ΘM n (ai , 1) = ∇Mn . (5) The only simple mono-unary algebras are cycles of prime order (exercise). The proof of the following theorem is also left as an exercise.
Ii) A ∼ = B × C implies either B or C is trivial. (iii) A has exactly two factor congruence relations, more precisely, the only two factor congruences of A are ∆A and ∇A . Proof. (i) =⇒ (ii): trivial (ii) =⇒ (iii). Let α and α ˆ be complementary congruences of A. Then A ∼ ˆ = A/α ×A/α. By assumption A/α or A/α ˆ is trivial. In the first case we have α = ∇A and hence ˆ =α∩α ˆ = ∆A . If A/α ˆ is trivial, then α ˆ = ∇A and α = ∆A . So ∆A and ∇A α ˆ = ∇A ∩ α are the only factor congruences of A. (iii) =⇒ (i).
Algebraists' Homage: Papers in Ring Theory and Related Topics by S. A. Amitsur, D. J. Saltman, George B. Seligman