Totally Principal Ideal Rings
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Given a commutative ring A, the ideals are the main tool to study its structure. One of the simplest cases is that in which every ideal is principal; thus we have principal ideal rings (PIRs) and principal ideal domains (PIDs), whose structure is well known. In the literature there are several generalisations of PIRs and PIDs; one of them is related to a multiplicative set S ⊆ A: Our main goal in this work is to go deeper into this generalisation from a more general point of view, which will allow us to establish and reach new results on the structure of these rings. The framework will be hereditary torsion theories that extend both the classical and relative S-case. We highlight the interest of the S–torsion ideals and the S–closed prime ideals as main tools, and introduce the totally S–PIRs. In particular, in addition to various characterizations of the totally σ–PIRs, we show that there exists a structure theorem for totally σ–PIRs, showing that they decompose, modulo the σ–torsion ideal, as a product of indecomposable σ–PIRs. We conclude the paper by showing several examples to illustrate the theory.
Abstract
Given a commutative ring A, the ideals are the main tool to study its structure. One of the simplest cases is that in which every ideal is principal; thus we have principal ideal rings (PIRs) and principal ideal domains (PIDs), whose structure is well known. In the literature there are several generalisations of PIRs and PIDs; one of them is related to a multiplicative set S ⊆ A: Our main goal in this work is to go deeper into this generalisation from a more general point of view, which will allow us to establish and reach new results on the structure of these rings. The framework will be hereditary torsion theories that extend both the classical and relative S-case. We highlight the interest of the S–torsion ideals and the S–closed prime ideals as main tools, and introduce the totally S–PIRs. In particular, in addition to various characterizations of the totally σ–PIRs, we show that there exists a structure theorem for totally σ–PIRs, showing that they decompose, modulo the σ–torsion ideal, as a product of indecomposable σ–PIRs. We conclude the paper by showing several examples to illustrate the theory.