Conventions

The image, kernel, and cokernel of a group homomorphism f:A—B will

be denoted as usual by Im(/), Ker(/), Coker(/), respectively. Thus Coker(/) =

B/Im(f). Given an abelian group A and a positive integer n, we denote the image,

kernel, and cokernel of the homomorphism A — A of multiplication by n by nA,

n

A,

and A/n, respectively.

For a prime number p we set 7LV — JimZ/p1. Likewise, we set Z = |imZ/n,

where n ranges over all positive integers, and the inverse limit is with respect to

the divisibility relation.

Unless explicitly stated otherwise, all rings will be tacitly assumed to be com-

mutative with 1, and all modules two-sided (an important exception will be the

^-structures, discussed in Part IV, which are anti-commutative rings). The group

of invertible elements in a ring R will be denoted by

Rx.

In particular, the mul-

tiplicative group F \ {0} of a field F will be denoted by Fx. A grading on a ring

will always be by the nonnegative integers.

Given a subset A of a group, we denote the subgroup it generates by (A). The

notation B A will mean that B is a subgroup of the group A.

Given a subsets A,B of a field F and an element c of F we set

A ± B = {a ± b | a G A, b G B}, AB = {ab \ a G A, b G B}

-A = {-a | a G A}, cA = {ca \ a G A},

etc.

We denote the fixed field of a group G of automorphisms of a field E by

EG.

If

a is an element of some field extension of E and is algebraic over E, then we denote

its irreducible polynomial over E by irr(a, E). An extension F C E of fields will be

written as E/F, and its transcendence degree will be denoted by tr.deg(E'/F).

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