By Warren Colburn

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26. Let F be a field, and V = F xi ⊂ F {X}. The Grassmann, or exterior, algebra is G := F {X}/I where I is the ideal v 2 : v ∈ V of F {X}. ) Since I is generated by even polynomials, we see that I ∩ V = 0, so we view V ⊆ G; from this point of view (which is the one we shall take), G is the algebra generated by V modulo the relations v 2 = 0, ∀v ∈ V. 27. Since we shall refer to the Grassmann algebra so often, let us fix some basic notation. We fix a base e1 , e2 , . . of V . 13) which clearly is equivalent to the following set of relations: e2i = 0; ei ej = −ej ei , ∀i = j.

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.

Gn,i ) for suitable polynomials g0,i , g1,i , g2,i , . . , gn,i . Proof. 23. (ii) ⇒ (iii) by comparing the different monomials of f . (iii) ⇒ (iv) is immediate. (iv) ⇒ (i) is clear, since cn is n-alternating. 30. Thus, by (iv), each n-alternating polynomial can be written in terms of cn . If cn ∈ id(A), then id(A) contains every n-alternating polynomial. 31. The alternator of a multilinear polynomial f in indeterminates xi1 , . . ,in ;X) := sgn(π)f, summed over all permutations π of {i1 , .

### An introduction to algebra: Upon the inductive method of instruction by Warren Colburn

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