Constantin Nǎstǎsescu

Dimensions of ring theory

1. Finiteness Conditions for Lattices.- 1.1. Lattices.- 1.2. Noetherian and Artinian Lattices.- 1.3. Lattices of Finite Length.- 1.4. Irreducible Elements in a Lattice.- 1.5. Goldie Dimension of a Modular Lattice.- 1.6. Goldie Dimension and Chain Conditions for Modular Lattices with Finite Group Actions.- 1.7. Complements and Pseudo-Complements.- 1.8. Semiatomic Lattices and Compactly Generated Lattices.- 1.9. Semiartinian Lattices.- 1.10. Indecomposable Elements in a Lattice.- 1.11. Exercises.- Bibliographical Comments to Chapter 1.- 2. Finiteness Conditions for Modules.- 2.1. Modules.- 2.2. The Lattice of Submodules of a Module.- 2.3. Noetherian and Artinian Modules.- 2.4. Modules of Finite Length.- 2.5. Semisimple Modules.- 2.6. Semisimple and Simple Artinian Rings.- 2.7. The Jacobson Radical and the Prime Radical of a Ring.- 2.8. Rings of Fractions. Goldies Theorems.- 2.9. Artinian Modules which are Noetherian.- 2.10. Projective and Infective Modules.- 2.11. Tensor Product and Flat Modules.- 2.12. Normalizing Extensions of a Ring.- 2.13. Graded Rings and Modules.- 2.14. Graded Rings and Modules of Type ?. Internal Homogenisation.- 2.15. Noetherian Modules over Graded Rings of Type ?. Applications.- 2.16. Strongly Graded Rings and Clifford Systems for Finite Groups.- 2.17. Invariants of a Finite Group Action.- 2.18. Exercises.- Bibliographical Comments to Chapter 2.- 3. Krull Dimension and Gabriel Dimension of an Ordered Set.- 3.1. Definitions and Basic Properties.- 3.2. The Krull Dimension of a Modular Lattice.- 3.3. Critical Composition Series of a Lattice.- 3.4. The Gabriel Dimension of a Modular Lattice.- 3.5. Comparison of Krull and Gabriel Dimension.- 3.6. Exercises.- Bibliographical Comments to Chapter 3.- 4. Krull Dimension and Gabriel Dimension of Rings and Modules.- 4.1. Definitions and Generalities.- 4.2. Krull and Gabriel Dimension of Some Special Classes of Rings and Modules.- 4.2.1. The Ring of Endomorphisms of a Projective Finitely Generated Module.- 4.2.2. Normalizing Extensions.- 4.2.3. Rings Strongly Graded by a Finite Group.- 4.2.4. The Ring of Invariants.- 4.2.5. Graded Rings of Type ?.- 4.2.6. Filtered Rings and Modules.- 4.2.7. Ore and Skew-Laurent Extensions.- 4.2.8. Affine P.I. Algebras.(Addendum).- 4.3. Exercises.- Bibliographical Comments to Chapter 4.- 5. Rings with Krull Dimension.- 5.1. Nil Ideals.- 5.2. Semiprime Rings with Krull Dimension.- 5.3. Classical Krull Dimension of a Ring.- 5.4. Associated prime Ideals.- 5.5. Fully Left Bounded Rings with Krull Dimension.- 5.6. Examples of Noetherian Rings of Arbitrary Krull Dimension.- 5.7. Exercises.- Bibliographical Comments to Chapter 5.- 6. Krull Dimension of Noetherian Rings. The Principal Ideal Theorem.- 6.1. Fully Left Bounded Left Noetherian Rings.- 6.2. The Reduced Rank of a Module.- 6.3. Noetherian Rings Satisfying Condition H.- 6.4. Fully Bounded Noetherian Rings.- 6.5. Krull Dimension and Invertible Ideals in a Noetherian Ring.- 6.6. The Principal Ideal Theorem.- 6.7. Exercises.- Bibliographical Comments to Chapter 6.- 7. Relative Krull and Gabriel Dimensions.- 7.1. Additive Topologies and Torsion Theories.- 7.2. The Lattices CF (M) and CHg.- 7.3. Relative Krull Dimension.- 7.4. Relative Krull Dimension Applied to the Principal Ideal Theorem.- 7.5. Relative Gabriel Dimension.- 7.6. Relative Krull and Gabriel Dimensions of Graded Rings.- 7.7. Exercises.- Bibliographical Comments to Chapter 7.- 8. Homological Dimensions.- 8.1. The Projective Dimension of a Module.- 8.2. Homological Dimension of Polynomial Rings and Rings of Formal Power Series.- 8.3. Injective Dimension of a Module.- 8.4. The Flat Dimension of a Module.- 8.5. The Artin-Rees Property and Homological Dimensions.- 8.6. Regular Local Rings.- 8.7. Exercises.- Bibliographical Comments to Chapter 8.- 9. Rings of Finite Global Dimension.- 9.1. The Zariski Topology.- 9.2. The Local Study of Homological Dimension.- 9.3. Rings Integral over their Centres.- 9.4. Commutative Rings of Finite Global Dimension.- 9.5. Exercises.- Bibliographical Comments to Chapter 9.- 10. The Gelfand-Kirillov Dimension.- 10.1. Definitions and Basic Properties.- 10.2. GK-dimension of Filtered and Graded Algebras.- 10.3. Applications to Special Classes of Rings.- 10.3.1. Rings of Differential Operators and Weyl Algebras.- 10.3.2. Remarks on Enveloping Algebras of Lie Algebras.(Addendum)..- 10.3.3. P.I.Algebras.(Addendum).- 10.4. Exercises.- Bibliographical Comments to Chapter 10.- References.

Torsion theories for coalgebras

Abstract For any coalgebra C, we can consider the category of all right C-comodules MC, which is an abelian category. We give a complete characterization of hereditary (pre)torsion theories in MC. In the first part of the paper we study monomorphism and epimorphism in the category of coalgebras, since they are related to the torsion theories determined by certain morphism of coalgebras with some additional properties (e.g. coflat morphisms).

Gradings of finite support. Application to injective objects

Let R be a group-graded ring. In this paper we study the relationship between injective objects in R-gr and in R-mod. It is well-known that gr-injectives, i.e. injective objects in R-gr, need not be injective in R-mod. However, in [lo], the second author showed that if R has finite support, then gr-injective modules with finite support are injective in R-mod. We generalize this result by showing that the restriction that R have finite support is unnecessary. Suppose M is a graded R-module with finite support, R also with finite support. In the first section we show that although sometimes one can grade M and R by a finite group, while preserving the homogeneous components of the grading, this is not always possible. Thus the theory of graded rings and modules with finite support does not coincide with the theory of finite group gradings. In Section 2, the full subcategory Cx of R-gr of graded R-modules with support in X s G is introduced. We show that the forgetful function Ux: Cx -+ R-mod, has a right adjoint, and if X is finite, also a left adjoint. We determine necessary and sufficient conditions for Cx to be equivalent to a module category. If X is finite, either Cx is zero or equivalent to be module category, S/I-mod, where S is Quinn’s smash product. Section 3 uses the category Cx to establish the main result, namely that gr-injectives with finite support are injective. As corollaries, we give necessary and sufficient conditions for an injective R-modules (injective indecomposable R-module) to be gradable if G is finite (supp R is finite). Finally, we show that if G is infinite, supp (R) finite and every gr-injective module is injective, then R is left noetherian, thus giving a converse to a result of the second author [lo].

Rings with Krull Dimension

The aim of this section is to establish that a nil ideal of a ring with Krull dimension is nilpotent.

Finiteness Conditions for Lattices

Let ≤ denote a partial ordering on a nonempty set L and let < be defined by: a < b with a,b ∈ L if and only if a ≤ b and a ≠ b. If a,b ∈ L and b ≤ a then a is said to contain b. If M is a subset of L, then an a ∈ L such that x ≤ a, resp. a ≤ x, for all x ∈ M is said to be an upper, resp. lower, bound for the set M. An element a ∈ L is said to be the supremum, resp. infemum, of M if a is the least upperbound, resp. the largest lower bound, for M (i.e. a is an upper (resp. lower) bound for M and if a’ is another upper (resp. lower) bound for M then we have a ≤ a’, resp. a’ ≤ a). A supremum, resp. infemum of M is unique and we will denote it by sup(M) or ⋁ x∈M x, resp. inf(M) or ⋀ x∈M x. When using the notation ⋁ x∈M x, resp. ⋀ x∈M x, we will also refer to these elements as the join, resp. meet, of the elements of M. In case M = {x1,..., x n }, we also write sup(M) = ⋁ i=1 n x i = x1 ⋁...⋁x n and inf(M) = ⋀ i=1 n x i = x1 ⋀ x2 ⋀...x n .

Krull Dimension and Gabriel Dimension of Rings and Modules

Let R be a ring, M a left R-module. We say that M has Krull dimension, resp. Gabriel dimension, if the lattice L(M) has Krull dimension, resp. Gabriel dimension. The ordinal Kdim L(M), resp. Gdim L(M), will be denoted by Kdim M, resp. Gdim M, and it is called the Krull dimension, resp. Gabriel dimension, of M.

Rings of Finite Global Dimension

In this chapter we study the connection between the Krull dimension and the homological dimension of a left Noetherian ring. Main results in this chapter will be Theorem 9.3.10. and 9.4.10. However the result of Theorem 9.3.10. may be strenghtened considerably (as we point out in the bibliographical comments at the end) but we did not include these more extended versions here because they require more techniques than what we are willing to introduce here.

Finiteness Conditions for Modules

In this book all rings considered are associative rings with unit unless otherwise stated. We write R for such a ring, 1 ∈ R is its unit element. A left R-module is an abelian group (the group law is denoted additively) with a scalar multiplication R × M → M, (a,x) ↦ ax satisfying the following properties: (1.) (a+b)x = ax + bx for x ∈ M, a, ∈ R (2.) a(x+y) = ax + ay for x,y ∈ M, a ∈ R (3.) (ab)x = a(bx) for x ∈ M, a,b ∈ R (4.) 1.x = x for x ∈ M

Krull Dimension of Noetherian Rings. The Principal Ideal Theorem

In this and consequent sections we study some properties of fully left bounded and fully (left and right) bounded rings.

Krull Dimension and Gabriel Dimension of an Ordered Set

For a partially ordered set (L,≤) we let Г(L) be the set {(a,b), a ≤ b, a,b ∈ L}. By transfinite recursion we may define on Г(L) a filtration in the following way: Г-1(L) = {(a,b) ∈ Г(L),a = b} Г0(L) = {(a,b) ∈ Г(L), [a,b] is Artinian} Гα(L) = {(a,b) ∈ Г(L), for all b ≥ b1 ≥… ≥ b n ≥… ≥ a, there is an n ∈ ℕ such that [bi+1, b i ) ∈ Uβ<α Г β (L), for each i ≥ n}, where we assume that for each ordinal β, β < α, Г β (L) is already defined. In this way we obtain an ascending chain Г-1(L) ⊂ Г0(L) ⊂… ⊂ Г α (L) ⊂.... Since L is a set it follows that Гς(L) = Гς+1(L) =…. If Г(L) = Г α (L) for some ordinal α then we say that the Krull dimension of L is defined.