Commutative ring

A commutative ring JT is called a field if, in addition to the properties given above, the following two postulates also hold true ... 

Conventional CA are defined by assigning to each of the N linearly connected sites, a time-dependent variable Oi t) (i=0,l,.,, N), belonging to a finite commutative ring TZk (usually represented by the integers modulo A , Zk). These site values, or colors, evolve iteratively according to a range-r mapping (j> ... 

A. R. Magid, The Separable Galois Theory of Commutative Rings (1974)... 

B. R. McDonald, Linear Algebra Over Commutative Rings (1984)... 

J. A. Huckaba, Commutative Rings with Zero Divisors (1988)... 

Vol. 1534 C. Greither, Cyclic Galois Extensions of Commutative Rings. X, 145 pages. 1992. 

The idea on Which this piart is based is an algebraic version of differentiation which will serve in all characteristics as a replacement for the differential part of real Lie group theory. The crucial feature turns out to be the product rule. Specifically, let A be a fc-algebra, M an A-module. A derivation Dot A into M is an additive map D A - M satisfying D(ab) = aD(b) + bD(a). We say D is a k-derivation if it is fc-linear, or equivalently if D(k) = 0. Ultimately k here will be a field, but for the first three sections it can be any commutative ring. 

W is a complete local ring of characteristic zero with maximal ideal pW, and residue class field W/pW = k these properties of W determine it uniquely (k being perfect), and o is the unique automorphism inducing the automorphism of raising to p-th powers in the residue class field (compare J.-P.Serre -Corps locaux, II.5 6), We denote by A the (non-commutative) ring generated over W by F and V with the relations... 

It is possible to associate a geometric object to an arbitrary commutative ring R. This object will be called Spec (R). If R is a finitely generated integral domain over an algebraically closed field, Spec (R) will be very nearly the same as an affine variety associated to R in Chapter I. However in this section we will be completely indifferent to any special properties that R may or may not have -e.g., whether R has nilpotents or other zero-divisors in it or not whether or not R has a large subfield over which it is finitely generated or even any subfield at all. We insist only that R be commutative and have a unit element 1. 

More generally, let R be any commutative ring with descending chain condition. Then R is the direct sum of its primary subrings ... 

UaiQ.x ua is isomorphic to (Spec (Ra), Pspec (i a)) or some commutative ring R. 

Corollary 1. The category of affine schemes is isomorphic to the category of commutative rings with unit, with arrows reversed. 

Next we discuss the sheafified version of the above. Let f/ be a topological space, O a sheaf of commutative rings, and A the abelian category of (sheaves of) C>-modules. The sheaf-hom functor... 

Let 7 be a topological space, O a sheaf of commutative rings, and A the abelian category of (sheaves of) O-modules. Recall from (1.5.4) the definition of the tensor product (over O) of two complexes in K(.A), and its A-functorial properties. The standard theory of the derived tensor product, via resolutions by complexes of flat modules, applies to complexes in D (.4), see e.g., [H, p.93], Following Spaltenstein [Sp] we can use direct limits to extend the theory to arbitrary complexes in D(.A). Before defining, in (2.5.7), the derived tensor product, we need to develop an appropriate acyclicity notion, q-flatness. ... 

Lemma 32.1. Let R he a noetherian commutative ring, and G a finite group which acts on R. Set A = R, and assume that Spec A is connected. Then G permutes the connected components of Spec R transitively. 

Let 7 be a small category, and B, a covariant functor from I to the category of (non-commutative) rings. A left R,-module is a collection M = (/ 0)0eMor(/)) such that M is a left Bj-module for each... 

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