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Many problems in commutative algebra treat various ways that a fixed ideal (or each ideal of a given class of ideals) of a commutative ring can be decomposed. Generally speaking, early problems of this type that arose from algebraic geometry concerned representations of ideals as intersections, while those arising from algebraic number theory involved representations in terms of products.

In Chapter 4 the following topics are considered: the radical of the endomorphism ring of a p-group (Section 20); the radical of the endomorphism ring of a torsion-free group of finite rank (Section 21); the radical of the endomorphism ring of an algebraically compact group and a completely decomposable torsion-free group (Section 22); the nilpotence of the radicals N(End(G)) and J(End(G)). (Section 23).

In Chapter 7 the following topics are considered: homogeneous fully transitive groups (Section 40); fully transitive groups whose quasi-endomorphism rings are division rings (Section 41); fully transitive groups coinciding with their pseudo-socles (Section 42); fully transitive groups with restrictions on element types (Section 43); torsion-free groups of p-ranks ≤ 1 (Section 44).

In Chapter 1 the following topics are considered: some general results about rings, modules, categories, and Abelian groups (Sections 1 and 2); examples and some properties of endomorphism rings (Section 3); torsion-free rings of finite rank (Section 4); the quasi-endomorphism ring of a torsion-free group (Section 5); E-modules and E-rings (Section 6); torsion-free groups coinciding with their pseudo-socles (Section 7); irreducible torsion-free groups (Section 8).

In Chapter 2 the following topics are considered: endo-Artinian and endo-Noetherian groups (Section 9); endo-flat primary groups (Section 10); endo-finite torsion-free groups of finite rank (Section 11); endo-projective and endo-generator torsion-free groups (Section 12); endo-flat torsion-free groups of finite rank (Section 13).

In Chapter 3 the following topics are considered: the finite topology (Section 14); endomorphism rings with the minimum condition (Section 15); Hom(A, B) as a Noetherian module over End(B) (Section 16); mixed groups with Noetherian endomorphism rings (Section 17); regular endomorphism rings (Section 18); commutative and local endomorphism rings (Section 19).

In Chapter 5 the following topics are considered: the Baer-Kaplansky theorem (Section 24); continuous and discrete isomorphisms of endomorphism rings (Section 25); endomorphism rings of groups with large divisible subgroups (Section 26); isomorphisms of endomorphism rings of mixed groups of torsion-free rank 1 (Section 27); the Corner’s theorem on split realization (Section 28); realizations for endomorphism rings of torsion-free groups (Section 29); the realization problem for endomorphism rings of mixed groups (Section 30).

In Chapter 6 the following topics are considered: self-small groups (Section 31); equivalences of categories of groups and categories of modules over endomorphism rings (Section 32); faithful groups (Section 33); faithful endo-fiat groups (Section 34); groups with right hereditary endomorphism rings (Section 35); groups of generalized rank 1 (Section 36); torsion-free groups with hereditary endomorphism rings (Section 37); maximal orders as endomorphism rings (Section 38); p-semisimple groups (Section 39).

Let R be a ring and let ∆ be a set of operators acting on R. In this case R is said to be a ∆-ring and ∆ is called its basic set of operators. In the following sections the operators in ∆ will be either derivation operators or endomorphisms but now we do not impose any restrictions on ∆.

In this chapter we deal with rings, fields and modules on which a finitely generated commutative group G acts. For such structures generalizations of the theorems on difference dimension polynomials and their invariants are derived. The main results of the chapter are Theorem 8.2.1 (which establishes the existence of dimension polynomial of an excellently filtered A-G-module over an artinian G-ring), Theorem 8.2.5 (this theorem describes the invariants of a dimension polynomial of a G-module over a G-field) and Theorem 8.3.16 (an analogue of Kolchin Theorem on differential dimension polynomial (see Theorem 5.4.1) that also generalizes Theorem 6.4.1 on difference dimension polynomial). The last result is helpful for the study of systems of “G-equations”, i.e., algebraic equations with respect to indeterminates and their images under the actions of elements of a group G. Using Theorem 8.3.16, for any such system, one can consider the appropriate dimension polynomial and use it to determine the “strength” of the system in the sense of A.Einstein [Ei53].

Let R be a difference ring with a basic set σ = {α l,...,α n } and let T = T σ be a free commutative semigroup generated by the elements α l,...,α n . As in Section 3.3, by the order of an element \(\tau = \alpha _1^{{k_1}}...\alpha _n^{{k_n}} \in T\left( {{k_1},...,{k_n} \in } \right)\) we shall mean the number ord \(\tau = \Sigma _{i = 1}^n{k_1}\) and set T r = {τ ∈ T | ord τ = r}, T(r) = {τ ∈ T | ord τ ≤ r} for any r ∈ ⑅. Furthermore, let U be a ring of difference (σ-) operators over the ring R. As in Chapter 3, if \(f = \sum\nolimits_{\tau \in T} {{a_\tau }\tau } \in U\) (a τ τ ∈ R for any τ ∈ T and a τ = 0 for almost all τ ∈ T), then the number ord f = max{ord τ | a τ ≠ 0} will be called the order of the element f.

Let R be a differential ring with a basic set ∆ = {d 1,..., d m }, D be the ring of linear differential operators over R (see Definition 3.2.38). As before, by T we denote the set of monomials of D (see Example 4.1.6 and Definition 4.1.4), and T(r) denotes the set of monomials whose order does not exceed r. Consider on D an ascending filtration (D r ) r∈ℤ, where D r = {f ∈ D | ord f ≤ r} = R ∙ T(r) for r ≥ 0, and D r = 0 for r < 0 (see Exercise 4.3.1). Below, if the contrary is not said explicitly, by a filtration on D we shall mean this filtration. By a filtered differential R-module we shall mean a D-module M with exhaustive and separable filtration (M r ) r∈ℤ. It means that M = ⋃ r∈ℤ M r and there exists r 0 ∈ ℤ such that M r = 0 for all r < r 0, M i ⊆ M i+1 and D i M r ⊆ M r+i for all r, i ∈ ℤ.

Let R be a commutative ring, M an R-module and U a family of R-submodules of M. Furthermore, let B U denote the set of all pairs (N, N′) ∈ U × U such that N ⊇ N′,and let \(\overline {\Bbb Z}\) be the set obtained by the adjoining of a new symbol ∞ to the set of integers ℤ. We shall consider \(\overline {\Bbb Z} \) as a linearly ordered set whose order < is the extension of the natural order of ℤ such that a < ∞ for all a < ℤ.

The first partial implementation by the authors of the described algorithms was made in 1980 [MP80]. The programs were written in the algorithmic language REFAL, run on the computer BESM-6 and were destinated to compute characteristic sets of differential ideals in the ring of differential polynomials.