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