Title:

Hereditary rings and rings of finite representation type

This thesis is a study of Noetherian PI rings with the property that every proper Artinian homomorphic image is of finite representation type. Our main result is: Theorem 5.1.12 Let R be a Noetherian PI ring, which is an order in an Artinian ring. Suppose that every proper Artinian factor ring of R is of finite representation type. Then R is a direct sum of an Artinian ring of finite representation type and prime hereditary rings. As a special case we have that a prime Noetherian PI ring is hereditary if and only if all its proper Artinian homomorphic images are of finite representation type. The thesis is organized as follows. In Chapter 1 we introduce the terminology and collect well known results on Noetherian rings that we shall use in later chapters. We also include some remarks on the relationship between the Jadic completion of a certain Noetherian prime PI ring and its centre that seem not appear in the literature. In Chapter 2 we present some known characterizations of hereditary rings. We introduce a discussion on rings of finite representation type and give motivation for our study. In Chapter 3 we begin the proof of our main theorem. We adapt the proof of a theorem by S. Brenner [10], which shows that the 2x2 upper triangular matrix ring T2(Z/p4Z) is of infinite type, to the more general case of T2(D/dnD), where D is a noncommutative local Dedekind prime PI ring and dD its maximal ideal. This result is crucial for the proof of Theorem 5.1.12. The reduction of our problem to this case is allowed by a theorem of M. Ausländer (cf. Theorem 3.2.1) on trivial extension rings of Artin algebras. We describe this theorem and show that it holds also for Artinian PI rings. Then we analyze the graph of links between maximal ideals of R. We show that if R is a prime ring satisfying the hypothesis of Theorem 5.1.12 then all cliques of maximal ideals of R are finite. In the last section of this chapter we look at connections between Artinian serial rings and rings of finite representation type. We show that if R is a semiperfect local Noetherian PI ring which is not Artinian and such that R/J(R)2 is of finite representation type, then R is a hereditary prime ring. Finally, we introduce some rings related to the ring R satisfying the hypothesis of Theorem 5.1.12, which inherit the property of the homomorphic images and that we shall use for the proof of the theorem. In Chapter 4 we prove Theorem 5.1.12 under the additional assumption that every clique of maximal ideals of R is finite. This is done by analyzing in detail the structure of the Jadic completion of the localisation of R at a clique of maximal ideals. Then from the results of Chapter 3 we deduce that Theorem 5.1.12 holds if R is semiprime. In Chapter 5 we prove that cliques of maximal ideals of R are indeed finite. This finishes the proof of our result. Further, we prove that a Noetherian PI ring whose proper Artinian homomorphic images are all serial is an order in an Artinian ring. In Chapter 6 we prove the analogue of Theorem 5.1.12 for a semiprime PI ring which is affine over a field. Then we give some examples of Noetherian PI rings of different global dimension to show that the assumption in Theorem 5.1.12 on the existence of an Artinian quotient ring for R is necessary. For completeness of our study, some results are stated and proved in more generality than it is needed for the proof of our main theorem.
