On a spanning subgraph of the annihilating-ideal graph of a commutative ring

The rings considered in this paper are commutative with identity which are not integral domains. Let [Formula: see text] be a ring. Let us denote the set of all annihilating ideals of [Formula: see text] by [Formula: see text] and [Formula: see text] by [Formula: see text]. With [Formula: see text], we associate an undirected graph, denoted by [Formula: see text], whose vertex set is [Formula: see text] and distinct vertices [Formula: see text] and [Formula: see text] are adjacent in this graph if and only if [Formula: see text] and [Formula: see text]. The aim of this paper is to study the interplay between the graph-theoretic properties of [Formula: see text] and the ring-theoretic properties of [Formula: see text].

Reverse Mathematics of Divisibility in Integral Domains

<p>This thesis establishes new results concerning the proof-theoretic strength of two classic theorems of Ring Theory relating to factorization in integral domains. The first theorem asserts that if every irreducible is a prime, then every element has at most one decomposition into irreducibles; the second states that well-foundedness of divisibility implies the existence of an irreducible factorization for each element. After introductions to the Algebra framework used and Reverse Mathematics, we show that the first theorem is provable in the base system of Second Order Arithmetic RCA0, while the other is equivalent over RCA0 to the system ACA0.</p>

Reverse Mathematics of Divisibility in Integral Domains

<p>This thesis establishes new results concerning the proof-theoretic strength of two classic theorems of Ring Theory relating to factorization in integral domains. The first theorem asserts that if every irreducible is a prime, then every element has at most one decomposition into irreducibles; the second states that well-foundedness of divisibility implies the existence of an irreducible factorization for each element. After introductions to the Algebra framework used and Reverse Mathematics, we show that the first theorem is provable in the base system of Second Order Arithmetic RCA0, while the other is equivalent over RCA0 to the system ACA0.</p>

Reverse mathematics of rings

<p>Using the tools of reverse mathematics in second-order arithmetic, as developed by Friedman, Simpson, and others, we determine the axioms necessary to develop various topics in commutative ring theory. Our main contributions to the field are as follows. We look at fundamental results concerning primary ideals and the radical of an ideal, concepts previously unstudied in reverse mathematics. Then we turn to a fine-grained analysis of four different definitions of Noetherian in the weak base system RCA_0 + Sigma-2 induction. Finally, we begin a systematic study of various types of integral domains: PIDs, UFDs and Bézout and GCD domains.</p>

Reverse mathematics of rings

<p>Using the tools of reverse mathematics in second-order arithmetic, as developed by Friedman, Simpson, and others, we determine the axioms necessary to develop various topics in commutative ring theory. Our main contributions to the field are as follows. We look at fundamental results concerning primary ideals and the radical of an ideal, concepts previously unstudied in reverse mathematics. Then we turn to a fine-grained analysis of four different definitions of Noetherian in the weak base system RCA_0 + Sigma-2 induction. Finally, we begin a systematic study of various types of integral domains: PIDs, UFDs and Bézout and GCD domains.</p>

Pairs of domains where all intermediate domains satisfy S-ACCP

The rings considered in this paper are commutative with identity. If [Formula: see text] is a subring of a ring [Formula: see text], then we assume that [Formula: see text] contains the identity element of [Formula: see text]. Let [Formula: see text] be a multiplicatively closed subset (m.c. subset) of a ring [Formula: see text]. An increasing sequence of ideals [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-stationary if there exist [Formula: see text] and [Formula: see text] such that [Formula: see text] for all [Formula: see text]. This paper is motivated by the research work [A. Hamed and H. Kim, On integral domains in which every ascending chain on principal ideals is [Formula: see text]-stationary, Bull. Korean Math. Soc. 57(5) (2020) 1215–1229]. Let [Formula: see text] be a m.c. subset of an integral domain [Formula: see text]. We say that [Formula: see text] satisfies [Formula: see text]-ACCP if every increasing sequence of principal ideals of [Formula: see text] is [Formula: see text]-stationary. Let [Formula: see text] be a subring of an integral domain [Formula: see text] and let [Formula: see text] be a m.c. subset of [Formula: see text]. We say that [Formula: see text] is an [Formula: see text]-ACCP pair if [Formula: see text] satisfies [Formula: see text]-ACCP for every subring [Formula: see text] of [Formula: see text] with [Formula: see text]. The aim of this paper is to provide some pairs of domains [Formula: see text] such that [Formula: see text] is an [Formula: see text]-ACCP pair, where [Formula: see text] is a m.c. subset of [Formula: see text].