- Order structures in algebra

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# Order structures in algebra

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Type

Thesis

Author

Nai, Yuan Ting

Supervisor

Zhao, Dongsheng

Abstract

In this thesis, some order structures derived from certain algebras, in particular, rings, are considered. The main focus is on “point-free” ring theory. We extend some results in algebra to the abstract setting; in particular, the various aspects of the lattice of all ideals of a commutative ring with identity and the multiplicative lattice as its natural abstraction are studied.

The first part of study is on principal elements in multiplicative lattices and their generalisations. The notion of “principal element” of a multiplicative lattice, which is intended to mimic the principal ideals of a commutative ring, has its restriction. We extend the notion of “principal element” to that of “principal mapping” between arbitrary posets. The principal elements of a given multiplicative lattice can be categorised as principal mappings from the given multiplicative lattice to itself.

In Chapter 3, we define and study weak principal mappings between posets, which generalise the notion of “weak principal element” in multiplicative lattices. We prove that a mapping, with an upper adjoint, between bounded modular lattices is principal if and only if it is weak principal.

In Chapters 4 and 5, we consider the relationship between subsets of a multiplicative lattice L and the open set lattices of the subspaces of the prime spectrum of L. One classical result is that the open set lattice of the prime spectrum of a commutative ring R, endowed with the hull kernel topology, is isomorphic to the lattice of all radical ideals of R. It is natural to consider the problem: For a given multiplicative lattice L and a subspace S of the prime spectrum of L, can we find a subset of L that is order isomorphic to the open set lattice of S? We prove that if L is a continuous multiplicative lattice, the open set lattice of the prime spectrum of L is isomorphic to the lattice of all m−semiprime elements of L. If L is a reduced mp-multiplicative lattice and an r-lattice in which every prime element is beneath a unique maximal element, then the lattice of all pure elements of L is isomorphic to the open set lattice of the subspace of all maximal elements of L. We also prove that the open set lattice of the minimal prime spectrum of a reduced coherent multiplicative lattice L, with non-zero annihilator for every minimal prime element of L, is isomorphic to the lattice of all normal elements of L.

Given a commutative ring R, several different types of rings can be constructed from R, such as ring of fractions and ring of polynomials. The lattices of all ideals of such new rings are closely linked to the lattice of all ideals of R. In Chapter 6, we investigate such links in the case of a ring and in the abstract setting. We characterize the subposets of Idl(R) which are isomorphic to Idl(S−1R) for some multiplicative closed subset S of R. We extend this to the case of an r−lattice L with S as a multiplicative closed subset of the set of the principal elements of L.

The first part of study is on principal elements in multiplicative lattices and their generalisations. The notion of “principal element” of a multiplicative lattice, which is intended to mimic the principal ideals of a commutative ring, has its restriction. We extend the notion of “principal element” to that of “principal mapping” between arbitrary posets. The principal elements of a given multiplicative lattice can be categorised as principal mappings from the given multiplicative lattice to itself.

In Chapter 3, we define and study weak principal mappings between posets, which generalise the notion of “weak principal element” in multiplicative lattices. We prove that a mapping, with an upper adjoint, between bounded modular lattices is principal if and only if it is weak principal.

In Chapters 4 and 5, we consider the relationship between subsets of a multiplicative lattice L and the open set lattices of the subspaces of the prime spectrum of L. One classical result is that the open set lattice of the prime spectrum of a commutative ring R, endowed with the hull kernel topology, is isomorphic to the lattice of all radical ideals of R. It is natural to consider the problem: For a given multiplicative lattice L and a subspace S of the prime spectrum of L, can we find a subset of L that is order isomorphic to the open set lattice of S? We prove that if L is a continuous multiplicative lattice, the open set lattice of the prime spectrum of L is isomorphic to the lattice of all m−semiprime elements of L. If L is a reduced mp-multiplicative lattice and an r-lattice in which every prime element is beneath a unique maximal element, then the lattice of all pure elements of L is isomorphic to the open set lattice of the subspace of all maximal elements of L. We also prove that the open set lattice of the minimal prime spectrum of a reduced coherent multiplicative lattice L, with non-zero annihilator for every minimal prime element of L, is isomorphic to the lattice of all normal elements of L.

Given a commutative ring R, several different types of rings can be constructed from R, such as ring of fractions and ring of polynomials. The lattices of all ideals of such new rings are closely linked to the lattice of all ideals of R. In Chapter 6, we investigate such links in the case of a ring and in the abstract setting. We characterize the subposets of Idl(R) which are isomorphic to Idl(S−1R) for some multiplicative closed subset S of R. We extend this to the case of an r−lattice L with S as a multiplicative closed subset of the set of the principal elements of L.

Date Issued

2017

Call Number

QA172.4 Nai

Date Submitted

2017