Dissertation
An analysis of additive behavior between multiplicative structures in matrix rings
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Summer 2020
DOI: 10.17077/etd.005536
Abstract
If $R$ is a ring, we call an element $x \in R$ clean if $x = e + u$ where $u \in U(R)$ and $e \in \text{idem}(R)$, and we also call a ring clean if every element is clean. Clean rings were first defined by W. K. Nicholson in 1977 as a tool to study exchange rings, and have since become their own subject of study. A new class of simple rings known as fine rings was defined by G. C\v alug\v areanu and T. Y. Lam is 2016. If $R$ is a ring, a nonzero element $x \in R$ is fine if $x = t + u$ where $u \in U(R)$ and $t \in \text{nil}(R),$ and $R$ is fine if every nonzero element in $R$ is fine. Additionally, a systemic study of $S$-rings, rings in which every element is a sum of a finite number of units, was proposed by R. Raphael in 1974. A common thread tying cleans rings, fine rings, and $S$-rings together is the decomposition of ring elements into sums of multiplicative structures.
If $p$ is prime, then the matrix ring $M_n(\mathds{Z}/p\mathds{Z})$ is clean with clean decomposition $M_n(\mathds{Z}/p\mathds{Z}) = GL_n(\mathds{Z}/p\mathds{Z}) + E_n(\mathds{Z}/p\mathds{Z})$ where $E_n(\mathds{Z}/p\mathds{Z})$ is the set of $\{0,1\}$-diagonal idempotents in $M_n(\mathds{Z}/p\mathds{Z}).$ We will use an exhaustive computer algorithm in order to optimize this decomposition for $M_2(\mathds{Z}/p\mathds{Z})$ and $M_3(\mathds{Z}/p\mathds{Z})$ where possible. For $M_2(\mathds{Z}/p\mathds{Z})$ we will also derive an optimal estimate.
We will also examine $M_n(R)$ when $R$ is an integral domain within the context of fine rings. We will minimize $\text{nil}(M_2(R))$ in the fine decomposition $M_2(R) \setminus \{0\} = \text{nil}(M_2(R)) + GL_2(R)$, and also derive an optimal estimate for $\text{nil}(M_n(R))$ where $n$ is any $n \in \mathds{N}.$
Finally, if $R$ is a ring, then we will define a set $A \subseteq R$ to be a unit supplement of $R$ if $R = A + U(R).$ We will show that if $R$ is a commutative Noetherian ring, then every unit supplement for $M_n(R)$ contains a finite unit supplement. Within this context, we will also discuss and produce results concerning the diagonal matrices, reduced echelon form matrices, and permutation matrices in $M_n(F)$ when $F$ is a field.
Details
- Title: Subtitle
- An analysis of additive behavior between multiplicative structures in matrix rings
- Creators
- Katie Burke
- Contributors
- Victor Camillo (Advisor)Miodrag Iovanov (Advisor)Maggy Tomova (Committee Member)Charles Frohman (Committee Member)Bruce Ayati (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Summer 2020
- DOI
- 10.17077/etd.005536
- Publisher
- University of Iowa
- Number of pages
- viii, 73 pages
- Copyright
- Copyright 2020 Katie Burke
- Language
- English
- Description bibliographic
- Includes bibliographical references (page 73).
- Public Abstract (ETD)
Matrices are rectangular arrays of mathematical objects, such as numbers, which are broadly used and studied throughout mathematics. Like the integers or real numbers, matrices may also be added and multiplied together. Consequently, individual matrices may possess certain multiplicative properties, such as invertible, idempotent, or nilpotent matrices. In certain situations, there are special sets of matrices in which each matrix is a sum of matrices with certain multiplicative properties. Some of these sets may be classified as fine rings, clean rings, or S-rings for example.
In this thesis, we will examine sets of matrices which fall under these categories. Specifically, we will seek to optimize these matrix sums by minimizing the number of matrices with special multiplicative properties needed in order to generate the entire set by using computer algorithms where possible, and deriving estimates otherwise.
- Academic Unit
- Mathematics
- Record Identifier
- 9983987895602771
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