By Euler L.

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1 The algebra of generalized polynomials . . 2 The relatively free product modulo a T -ideal . . . . . . 1 The grading on the free productfree product and relatively free product . . . . Exercises for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . 49 52 54 55 55 57 57 59 60 61 62 63 63 65 65 66 67 67 68 In this chapter, we introduce PI-algebras and review some well-known results and techniques, most of which are associated with the structure theory of algebras.

1 The free associative algebra . . . . . . . . . . . . . . . 2 Polynomial identities . . . . . . . . . . . . . . . . . . 3 Multilinearization . . . . . . . . . . . . . . . . . . . . 4 PI-equivalence . . . . . . . . . . . . . . . . . . . . . . 5 Alternating polynomials . . . . . . . . . . . . . . . . . 1 Capelli polynomials . . . . . . . . . . . . 6 The action of the symmetric group .

Dm f (a1 ⊗ k1 , . . , am ⊗ km ) = f (a1 , . . , am ) ⊗ k1d1 . . km . Thus one side is 0 iff the other side is 0. Hence id(A ⊗ K) ⊆ id(A). 29], for details. 19 holds over any infinite field. 20. If I, J ⊳A with I ∩J = 0, then A ∼PI A/I × A/J . ) Thus, by induction, if A is a subdirect product of A1 , . . , Am then A ∼PI A1 × · · · × Am . 21. d. d. algebras A1 and A2 with id(Aj ) ⊃ id(A) and id(A1 ) ∩ id(A2 ) = id(A). d. algebras, each satisfying “more” identities than A. 22. An n-linear polynomial f (x1 , .

### An inquiry into whether or not 1000009 is a prime number by Euler L.

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