Some Aspects of Ring Theory

The lectures given in the 1965 Summer meeting of the C.I. M.E. have been an attempt to summarize and survey the development of the theory of polynomial identities since they first appeared in a paper byDehn (19229 on Desarguian Geometries till their recent application to Geometry (1965) - giving an almost complete solution to the problem which arose from the paper of Dehn on Desarguian Non-Pappian Geometries.

But the survey is for from be being complete; applications to grqup representations, Jacobsons' rings the Kurosh problem and other aspects of the theory are missing in particular, it lacks completely-references. Some of the results appear in the book “Structure of Ring” by N. Jacobson and in the Lecture Notes on rings given by I.N. Herstein at the University of Chicago. Other results appear in various papers by Amitsur, Herstein, Kaplansky, Levitzki, Posner, Shyrshov and others Many recent extensions, in particular the results on polynomial identities with coefficients in arbitrary domains with appear in forthcoming papers by the author.

In this chapter we shall make a study of rings satisfying certain ascending chain conditions. In the non-commutative case-and this is really the only case with which we shall be concerned- the decisive and incisive results are three theorems due to Goldie. The main part of the chapter will be taken up with a presentation of these.

Definition. An element a in the ring R is regular if it is neither a left nor right zero divisor in R.

Multiplication Representations in Classes of Algebras Defined by Identities.

In this chapter we develop the basic concepts of representation theory for an arbitrary class of algebras defined by identities. If f is an element of a free non-associative algebra over a field Φ then we say that an algebra r/I satisfies the identity f = 0 if f is mapped into 0 by every homomorphism of the free algebra into r. If S is a subset of a free non-associative algebra then we denote by C(S) the class of algebras satisfying every identity f = 0, f ϵ S. The representation theory for C(S) has as its starting point the notion of an S-bimodule for an r in the class C(S). This is a vector space m/I with bilinear compositions (a, u) → au, (a, u) → ua of ((r,m) into m such that the algebra ϵ = = r+m;, with multiplication (a₁ +u₁)(a₂ +u₂) = a₁ a₂ + a₁ u₂ + u₁ a₂, a_i υr, u_i υm is in the class C(S). We can derive the explicit conditions on au and ua for an S-bimodule of r from the set S of defining identities. Moreover, these conditions can be expressed as conditions on the linear transformations u →au, u →ua in m and this leads to the notion of an S-multiplication representation (S-birepresentation) of r in the associative algebra Horn I (m m). It is convenient to generalize this concept to that of an S-multiplication specialization in which Horn (m, m) is replaced by an arbitrary associative algebra with an identity element 1. This leads to the notion of a universal S-multiplication envelope for r in C(S). The determination of such envelopes is one of the basic problems of the representation theory since the S-bimodules and S-multiplication representations for can be identified with right modules and representations of the associative universal envelope.

In their basic treatise [4], Cartan and Eilenberg treat homological dimension as a peripheral aspect of the general theory being developed, and so do the later books by Northcott [13] and Mac Lane[9]. In the meantime a more or less self-contained theory of homological dimension has come into being. A good account of the portions relevant for local rings appears in the book of Nagata [l2].

In these lectures I will “revisit” the account given in my mimeographed notes of 1959 [5], and I will add further relevant material on R-sequences and unique factorization. Where there is some novelty, proofs will be sketched.

In this article we shall discuss a certain homological tool, the Koszul complex, which relates two concepts important in local ring theory, namely depth and multiplicity.

We recall that a local ring R is a commutative, notherian ring with identity, having a unique maximal ideal, m,. The dimension of the local ring R is the longest integer d for which a strictly descending chain of prime ideals, m = J₀ ⊃ J₁⊃…⊃ J₁, of length d exists. Since R is noetherian, all ideals of R are finitely generated. In particular, m, is finitely generated, and according to Krull's principal ideal theorem, the number of elements required to generate m, is always greater than or equal to. dim R (the dimension of R). If m can be generated by precisely d= dim R elements, R is said to be a regular local ring. An ideal r of R is said to be an ideal of definition or m-primary if r contains some power of m This is equivalent to saying that R/r is an R-module of finite length. A set of elements x₁,…, x_d of R (where d = dim R) is said to be a system of parameters if the elements generate an ideal of definition.

The familiar Euclidean algorithm, for integers or polynomials over a field, depends on the division with quotient and remainder, which can be built up from the following basic step (for the case of a polynomial ring R) :

A. given a,b ∈ R, if ∂a ≥ ∂b ≥ 0, then there exists c ∈ R such that ∂(a-bc) < ∂a.

I. This is a summary only of the seminar given, a full account will appear in the Journal of Algebra.

This is a filtered set of right ideals F such that