Hidetoshi Marubayashi

Non-commutative valuation rings and semi-hereditary orders

Preface. I. Semi-Hereditary and Prufer Orders. II. Dubrovin Valuation Rings. III. Semi-Local Bezout Orders. IV. The Applications and Examples. Appendix: A1. Semi-Perfect Rings and Serial Rings. A2. Coherent Rings. A3. Azumaya Algebras. A4. The Lifting Idempotents. A5. Wedderburns Theorem. References. Index of Notation. Index.

A classification of prime segments in simple artinian rings

Let A be a simple artinian ring. A valuation ring of A is a Bezout order R of A so that R/J(R) is simple artinian, a Goldie prime is a prime ideal P of R so that R/P is Goldie, and a prime segment of A is a pair of neighbouring Goldie primes of R. A prime segment P1 ⊃ P2 is archimedean if K(P1) = {a ∈ P1|P1aP1 ⊂ P1} is equal to P1, it is simple if K(P1) = P2 and it is exceptional if P1 ⊃ K(P1) ⊃ P2. In this last case, K(P1) is a prime ideal of R so that R/K(P1) is not Goldie. Using the group of divisorial ideals, these results are applied to classify rank one valuation rings according to the structure of their ideal lattices. The exceptional case splits further into infinitely many cases depending on the minimal n so that K(P1) is not divisorial for n ≥ 2.

Partial Skew Polynomial Rings and Goldie Rings

We prove that if R is a semiprime ring and α is a partial action of an infinite cyclic group on R, then R is right Goldie if and only if R[x; α] is right Goldie if and only if R⟨x; α⟩ is right Goldie, where R[x; α] (R⟨x; α⟩) denotes the partial skew (Laurent) polynomial ring over R. In addition, R⟨x; α⟩ is semiprime while R[x; α] is not necessarily semiprime.

Prime factor rings of skew polynomial rings over a commutative Dedekind domain

This paper is concerned with prime factor rings of a skew polynomial ring over a commutative Dedekind domain. Let P be a non-zero prime ideal of a skew polynomial ring R = D[x; ], where D is a commutative Dedekind domain and is an automorphism of D. If P is not a minimal prime ideal of R, then R/P is a simple Artinian ring. If P is a minimal prime ideal of R, then there are two dierent types of P, namely, either P = p[x; ] or P = P 0 \ R, where p is a -prime ideal of D, P 0 is a prime ideal of K[x; ] and K is the quotient field of D. In the first case R/P is a hereditary prime ring and in the second case, it is shown that R/P is a hereditary prime ring if and only if M 2 + P for any maximal ideal M of R. We give some examples of minimal prime ideals such that the factor rings are not hereditary or hereditary or Dedekind, respectively.

Primes of Gauss Extensions Over Crossed Product Algebras

Let K be a skew field with total subring V and G be a right ordered group with cone P, so that the crossed product algebra K*G has a skew field D of fractions. We consider total subrings R of D with R ∩ K = V, describe the overrings in D, as well as subrings of R. For particular extensions R of V we determine the prime ideals of R in terms of prime ideals of V and prime ideals of overcones of P in G.

Partial Skew Polynomial Rings Over Semisimple Artinian Rings

Let R be a semisimple Artinian ring with a partial action α of ℤ on R, and let R[x; α] be the partial skew polynomial ring. By the classification of the set E of all minimal central idempotents in R into three different types, a complete description of the prime radical of R[x; α] is given. Moreover, it is shown that any nonzero prime ideal of R[x; α] is maximal and is either principal or idempotent. In the case where α is of finite type, it is shown that R[x; α] is a semiprime hereditary ring.

Multiplicative Ideal Theory in Noncommutative Rings

The aim of this paper is to survey noncommutative rings from the viewpoint of multiplicative ideal theory. The main classes of rings considered are maximal orders, Krull orders (rings), unique factorization rings, generalized Dedekind prime rings, and hereditary Noetherian prime rings . We report on the description of reflexive ideals in Ore extensions and Rees rings. Further we give necessary and sufficient conditions (or sufficient conditions) for well-known classes of rings to be maximal orders, and we propose a polynomial-type generalization of hereditary Noetherian prime rings.

Polynomials determining hereditary prime PI-rings

A ring is called left (resp. right) hereditary if every left (resp. right) ideal is projective. A Dedekind domain is a commutative domain which is hereditary.

Projective ideals of skew polynomial rings over HNP rings

ABSTRACT Let R be an hereditary Noetherian prime ring (or, HNP-ring, for short), and let S = R[x;σ] be a skew polynomial ring over R with σ being an automorphism on R. The aim of the paper is to describe completely the structure of right projective ideals of R[x;σ] where R is an HNP-ring and to obtain that any right projective ideal of S is of the form X𝔟[x;σ], where X is an invertible ideal of S and 𝔟 is a σ-invariant eventually idempotent ideal of R.

Crossed Product Orders Over Valuation Rings II: Tamely Ramified Crossed Product Algebras

Let V be a commutative valuation domain of arbitrary K rull-dimension, with quotient field F, let K be a finite Galois extension of F with group G, and let S be the integral closure of V in K. Suppose that one has a 2-cocycle on G that takes values in the group of units of S. Then one can form the crossed product of G over S, S * G, which is a V-order in the central simple F-algebra K * G. If S * G is assumed to be a Dubrovin valuation ring of K * G, then the main result of this paper is that, given a suitable definition of tameness for central simple algebras, K * G is tamely ramified and defectless over F if and only if K is tamely ramified and defectless over F. The residue structure of S * G is also considered in the paper, as well as its behaviour upon passage to Henselization.