The weak global dimension of Gaussian rings

Abstract

A Prüfer domain is defined as an integral domain for which each nonzero finitely generated ideal is invertible or, equivalently, projective. In general, if R is a domain, we have the following equivalent statements: (i) R is semihereditary (see §2.6 for the definition). (ii) w.gl.dim(R) ≤ 1 (see §2.5 for the definition). (iii) R_m is a chain domain for every 𝔪 ∈ Max(R) (see §1.2, §1.3 for definitions). (iv) R is a Gaussian domain (see §3.1 for the definition). (v) R is a Prüfer domain. The definition of each class of ring featured in (i)–(v) can be extended to commutative rings which are not necessarily integral domains, i.e., which may have zero-divisors. However, there are examples showing that no two of (i)–(v) are equivalent in this more general setting. In fact we have strict implications R is semihereditary ⇒ w.gl.dim(R) ≤ 1 ⇒ R_m is a chain ring for every maximal ideal 𝔪 of R ⇒ R is a Gaussian ring ⇒ (v) R is a Prüfer ring. We concentrate our studies on the weak global dimension of Gaussian rings. The authors of [BaGl] proposed a conjecture that there were only three possibilities for the weak global dimension of a Gaussian ring, namely 0, 1 or ∞. We follow the authors of [DT] by referring to this as the Bazzoni–Glaz Conjecture. In [Gl2, Theorem 2.2], it is shown that if the Gaussian ring R is a reduced ring (i.e., R is a ring for which the zero element is the only nilpotent element) then R has weak global dimension at most 1, verifying the Conjecture in this case. The case of non-reduced Gaussian rings is given a great deal of attention in the 2011 preprint [DT]. The authors of [DT] prove the Conjecture is true using a number of concepts from homological algebra. We give details for some of these results and refer to articles and books for others. The proof in [DT] is quite long, involving several reduction steps to reach the final outcome. In the Chapter 1 we introduce some well-established results from the ideal theory of commutative rings. For the most part, we will skip the proofs of these results and give references for them to the reader. This chapter is very important for the following chapters. We then look at some homological algebra definitions, results and methods in Chapter 2. In particular, this chapter will look at the weak global dimension of a ring R. The third chapter concentrates on the ideal structure of Gaussian rings, especially local Gaussian rings, detailing general properties of their internal structure. Finally, in Chapter 4, we give a detailed proof of the Bazzoni–Glaz Conjecture. This final lengthy chapter considers an all-inclusive number of cases of Gaussian rings and shows that the conjecture holds for every case.