Extension of Antoine’s Ring Construction

Tai Keun Kwak

doi:10.1155/2019/2720810

Abstract

This article concerns a kind of extension of Antoine’s ring construction which is unit-IFP and Armendariz. Such extensions are also shown to be unit-IFP and Armendariz, by which we may extend the classes of unit-IFP and Armendariz rings.

Highlights

Antoine constructed a kind of ring coproduct and proved that such a ring is Armendariz in [1]
Let R K〈S〉/I, R1 K〈S0〉 and R′2 N ∗ (R2) K〈T〉/J. en, we have the following: (1) R is a unit-IFP ring with U(R) 􏼈k + g + cδ ε | k ∈ K\{0}, g ∈ R′2, c ∈ R2TpR2, δ ∈ R, ε ∈ R2TqR2 for some p, q ≥ 1 with p + q ≥ n}, where R′2 is the subalgebra of R2 of all polynomials of zero constant term
C ∈ R2TpR2, ε ∈ R2TqR2􏼛, where R′2 is the subalgebra of R2 of all polynomials of zero constant term, and p, q ≥ 1, p + q ≥ n

Summary

Introduction

Antoine constructed a kind of ring coproduct and proved that such a ring is Armendariz in [1]. Kim et al showed that such an Armendariz ring is unit-IFP [2]. We consider an extended situation of Antoine’s construction and show that such construction provides examples of unit-IFP rings and Armendariz rings. E group of units in R is denoted by U(R). E polynomial ring with an indeterminate x over R is denoted by R[x], and let Cf(x) denote the set of all coefficients of given a polynomial f(x). Use J(R), N ∗ (R), N ∗ (R), and N(R) to denote the Jacobson radical, the prime radical, the upper nilradical (i.e., sum of all nil ideals), and the set of all nilpotent elements in R, respectively. It is well known that N ∗ (R) ⊆ N ∗ (R) ⊆ N(R) and N ∗ (R) ⊆ J(R)

Unit-IFP Property

Armendariz Property