描述
开 本: 16开纸 张: 胶版纸包 装: 平装是否套装: 否国际标准书号ISBN: 9787030424105
三年以后，Glaz又写了一篇关于交换的凝聚环的综述报告，介绍了凝聚环在交换代数中的地位。与此同时，关于非交换的凝聚环的研究也逐渐活跃起来。这正是本书的所要着重讨论的内容之一。
Notations
Chapter 1 A glance in rings and modules
1.1 Rings and modules
1.2 Complexes, homological dimensions and functors
1.3 Finitely generated and finitely presented modules
1.4 FP-injective and flat modules
Exercise
Chapter 2 Coherent rings
2.1 Definition and examples
2.2 Characterizations of coherent rings
2.3 Extensions of coherent rings
2.4 Some generalizations
Exercise
Chapter 3 FP-injective rings
3.1 Definition and examples
3.2 Characterizations of FP-injective rings
3.3 Extensions of FP-injective rings
3.4 FP-injective and QF rings
3.5 FC rings
Exercise
Chapter 4 Homological dimensions
4.1 FP-injective dimension
4.2 n-FC rings
4.3 Weak global dimension
4.4 Semihereditary rings
Exercise
Chapter 5 Some applications
5.1 Flat envelopes and FP-injective covers
5.2 Gorenstein flat modules
5.3 Gorenstein FP-injective modules
5.4 Gorenstein flat complexes
5.5 Gorenstein FP-injective complexes
5.6 Relative and Tate homology
Exercise
Appendix A Open questions
Appendix B Categories and fuctors
Appendix C Categories of complexes of modules
References
Index
In this preliminary chapter, we brie.y present some fundamental notions and related results which will be frequently used in the sequel.
1.1 Rings and modules
Modern ring theory rose from early 1900s. As a common generalization of some “concrete” algebraic structures with operations generalizing the arithmetic opera-tions of addition and multiplication, the notion of rings is widely involved in many branches of mathematics. One e.cient approach to investigate a ring, among others, is to study modules over it.
We assume that the reader has been acquainted with monoids and (abelian) groups.
1.1.1 Rings, homomorphisms and ideals
De.nition 1.1.1 Let R be a nonempty set with two binary operations, addition (+) and multiplication (·), such that (R, +) is an abelian group and (R, ·) is a monoid. As usual, the zero element of (R, +) is written as 0, while the identity element of (R, ·) is denoted by 1. The system (R, +, ·, 0, 1) is called an associative ring if the multiplication is distributive over the addition, i.e.,
a · (b + c)= a · b + a · c and (b + c) · a = b · a + c · a
for all a, b, c ∈ R.
Given a ring (R, +, ·, 0, 1) and a subset S of R, if 0, 1 ∈ S and (S, +, ·, 0, 1) is also a ring then we say (S, +, ·, 0, 1) is a subring of R.
We usually abbreviate (R, +, ·, 0, 1) to R and write a · b as ab in R if it does not cause any confusion. The element 1 is called the identity of R. Sometimes we use 1R to emphasize the identity 1 in a ring R especially in case there are more than one ring involved at the same time.
A ring R is said to be commutative provided (R, ·, 1) is a commutative monoid.
By a division ring we mean a ring R in which every nonzero element a is invertible in the sense that ab = ba = 1 for some b ∈ R. It is easy to see that such an element
.1
b is unique if it exists. In this case, we write b = aand call it the inverse of a. An invertible element in a ring is also called a unit.
If R is a ring in which ab 0 for all nonzero elements a and b, then we say
= R is a domain. A commutative domain is also known as an integral domain.A commutative division ring is called a .eld.
Example 1.1.2 Typical examples of commutative rings include Z, Q, R and C. In addition, let R[x] be the set of all polynomials in x with real coe.cients. Then R[x] is also a commutative ring with the usual addition and multiplication of polynomials. All these ring are integral domains. In particular, Q, R and C are .elds.
Example 1.1.3 In linear algebra, we encounter noncommutative rings. For in-stance, let Mn(R) be the set of all n × n matrices over R, where n is an integer greater than 1. Then Mn(R) is a noncommutative ring with the usual addition and multiplication of matrices.
The notion of matrix rings can be generalized to de.ne rings of some in.nite matrices as follows.
Let Γ be an in.nite set and f :Γ × Γ → R be a map, where R is a ring. Suppose f(α, β)= aαβ for each (α, β) ∈ Γ × Γ. Then we may write f in a matrix form
[aαβ]Γ×Γ (or simply [aαβ])
and call it a Γ × Γ matrix over R. The set of all Γ × Γ matrices over R shall be denoted by MΓ(R). Then for any f =[aαβ ] and g =[bαβ] ∈ MΓ(R) we shall de.ne the sum of f and g as usual. Thus
[aαβ]+[bαβ]=[cαβ],
where cαβ = f(α, β)+ g(α, β). But the usual product of matrices in Mn(R) is not always adaptable for matrices in MΓ(R). And associativity (xy)z = x(yz) can fail for in.nite matrices x, y, z even when all products concerned make sense. For instance, let
. 0 0 00 ··· .
.111 ··· .. 1 .100 ··· .
.1 0 00 ···
011 ··· 01 .10 ···
.1 .1 00 ···
x = ,y = ,z =
001 ··· 001 .1 ···
.
.
.
.
.1 .1 .10 ···
… ….
. … .. …. ..
.
. . ..
… …. ..
. . ..
. . ..
and adopt the usual product of matrices formally, then we have xy =1= yx and yz = 1 but x
= z [38, Example 3]. Fortunately, MΓ(R) has some subsets in which the usual product of matrices is adaptable.
Example 1.1.4 By a row .nite matrix in MΓ(R) we mean a matrix [aαβ] ∈ MΓ(R) such that, for each α ∈ Γ, there are at most .nitely many β with aαβ = 0. A matrix [aαβ] is said to be column .nite provided the transpose of [aαβ] (denoted by [aαβ]T) is row .nite. Now let A =[aαβ] and B =[bαβ] be two row .nite matrices in MΓ(R). We can de.ne the product of A and B as usual, that is, AB =[cαβ], where cαβ =三γ∈Γ aαγ bγβ . Thus, RFMΓ(R), the set of all row .nite matrices in MΓ(R) becomes a ring with respect to the operations de.ned above. Similarly, the column .nite matrices in MΓ(R) form a ring, which shall be denoted by CFMΓ(R).
Example 1.1.5 The best known example of a noncommutative division ring is
H = {a + bi + cj + dk | a, b, c, d ∈ R},
the ring of quaternions .rst described by the Irish mathematician William Rowan Hamilton in 1843. The addition in H is similar to that in C. More precisely, the sum of a+bi+cj+dk and a’ +b’i+c’j+d’k is (a+a’)+(b+b’)i+(c+c’)j+(d+d’)k. The multiplication in H is determined by the distributive law and the products of the base eleme
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