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By Mal'tsev N., Kuz'min E. N.

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Dn ). On the other hand, F [Λ] is N-graded when we take F [Λ]d to be spanned by all monomials of total degree d. This is a standard tool in the theory of commutative algebra, relating to Hilbert Series which we consider more generally for PI-algebras in Chapter 11. (ii) In analogy to (i), the free algebra C{X} is N-graded, seen by taking C{X}n to be the homogeneous polynomials of degree n. This has already been used implicitly (in treating homogeneous polynomials) and enables us to specify many of their important properties.

2 Nonrepresentable algebras . . . . . . . . . . . . . . . . 1 Bergman’s example . . . . . . . . . . . . 3 Representability of affine Noetherian PI-algebras . . . . 4 Nil subalgebras of a representable algebra . . . . . . . 7 Sets of Identities . . . . . . . . . . . . . . . . . . . . . . . . . 1 The set of identities of an algebra . . . . . . . . . . . . 2 T -ideals and related notions . . . .

12. 5 9 Central localization The localization procedure can be generalized directly from the commutative situation to S −1 A whenever S is a (multiplicative) submonoid of Cent(A). In particular the ideals of S −1 A are precisely those subsets S −1 I where I ⊳ A. We say that an element s ∈ A is regular when sa, as = 0 for all a = 0 in A. When A is prime, then every submonoid of Cent(A) is regular. Here is an easy but useful result. 17. Suppose S is a submonoid of Cent(A) which is regular in A. Then S −1 A is prime iff A is prime.