A representation theorem for Chain rings

Abstract

Abstract.A ring A is called a chain ring if it is a local, both sided artinian, principal ideal ring. Let R be a commutative chain ring. Let A be a faithful R-algebra which is a chain ring such that A = A/J(A) is a separable field extension of R = R/J(R). It follows from a recent result by Alkhamees and Singh that A has a commutative R-subalgebra R 0 which is a chain ring such thatThe structure of A in terms of a skew polynomial ring over R 0 is determined. Let R be a commutative chain ring, and A be a local ring that is a faithful R-algebra. Then J(R) = R ∩ J(A). Let A = A/J(A) be a separable, algebraic field extension of R, and let A be either a locally finite Ralgebra or an artinian duo ring. As proved in [1], A has a commutative local R-subalgebra R 0 such thatThis subalgebra R 0 is also called a coefficient subring of A; such a subring is a commutative chain ring, and is a faithful R-algebra. The group of R-automorphisms of R 0 is investigated in Section 2. Wirt [8] introduced the concept of a distinguished basis of a bimodule over a Galois ring. In Section 3 an analogous concept for bimodules over R 0 is investigated.The main purpose of this paper is to prove a representation theorem for A, in case A is a chain ring, in terms of an appropriate homomorphic image of a skew polynomial ring over its coefficient subring. Sections 4 and 5 are devoted to proving the main theorem (Theorem 5.5). By Cohen [5], any

Automorphisms of a chain ring

“…(iii) Suppose k ≤ k. If θ k = 0, the result follows from the remarks above[2, Lemma 5.1]. If θ k = 0, the result follows from [2, Theorem 5.5].…”

“…For the definition and structure theorems on Galois rings one may refer to [6]. The following result is given by Lemma 2.3 and Theorem 2.7 in [2]. …”

“…There exists a least positive integer k such that θ k = πw for some unit w ∈ A, m = k(n − 1) + t for some 1 ≤ t ≤ k, A = R 0 ⊕ R 0 θ + · · · + R 0 θ k−1 . Further, as left R 0 -modules, R 0 ∼ = R 0 θ i for 0 ≤ i < t and R 0 θ i ∼ = R 0 /R 0 π for t ≤ i ≤ k − 1 [2]. Let k be the order of σ.…”

“…If A is commutative, σ = i R 0 , so k = 1; in this case, we say that (θ, π, m, n, k, w) is an associated set of A. Any unit b ∈ A is said to be lift algebraic over R, it there exists a monic polynomial f (x) ∈ R [x], which is irreducible modulo J (R) and is such that f (b) = 0 [1]; this polynomial is called an associated polynomial of b [2]. For any automorphism λ of a ring T , and any subset X of T , X λ denotes the set {a ∈ X : λ(a) = a}, in particular, T λ denotes the fixed ring of λ.…”