Are you preparing for a ring theory interview?
This guide is your ultimate resource to help you ace your interview and showcase your expertise in this fascinating field of abstract algebra. We’ve compiled a comprehensive list of ring theory interview questions gleaned from top sources like InterviewPrep and Math Stack Exchange, to equip you with the knowledge and confidence you need to succeed.
What is Ring Theory?
Ring theory is a branch of abstract algebra that studies rings, which are algebraic structures consisting of a set of elements with two binary operations, addition and multiplication, satisfying specific properties. Rings are fundamental to various mathematical disciplines, including number theory, algebraic geometry, and topology
Why is Ring Theory Important?
Understanding ring theory is crucial for mathematicians and computer scientists working in various fields. It provides a framework for studying algebraic structures, including fields, integral domains, and modules, which are essential for understanding abstract concepts like symmetry, groups, and vector spaces.
What to Expect in a Ring Theory Interview?
Ring theory interview questions can range from basic definitions and concepts to more complex problems and applications. Be prepared to demonstrate your understanding of key topics like:
- Ring definitions and properties
- Ideals, subrings, and quotient rings
- Homomorphisms and isomorphisms
- Polynomial rings and modules
- Noetherian rings and Hilbert’s Basis Theorem
- Applications of ring theory in other areas of mathematics
Let’s Dive into the Ring Theory Interview Questions!
1. What is a ring?
A ring is an algebraic structure $(R, +, cdot)$ consisting of a set $R$ with two binary operations, addition (+) and multiplication (·), satisfying the following properties:
- Closure under addition and multiplication: For any $a, b in R$, $a + b in R$ and $a cdot b in R$.
- Associativity of addition and multiplication: For any $a, b, c in R$, $(a + b) + c = a + (b + c)$ and $(a cdot b) cdot c = a cdot (b cdot c)$.
- Commutativity of addition: For any $a, b in R$, $a + b = b + a$.
- Existence of additive identity: There exists an element $0 in R$ such that for any $a in R$, $a + 0 = a$.
- Existence of additive inverse: For any $a in R$, there exists an element $-a in R$ such that $a + (-a) = 0$.
- Distributivity of multiplication over addition: For any $a, b, c in R$, $a cdot (b + c) = (a cdot b) + (a cdot c)$ and $(b + c) cdot a = (b cdot a) + (c cdot a)$.
2. What is an ideal?
An ideal $I$ of a ring $R$ is a non-empty subset of $R$ that satisfies the following properties:
- Closure under addition: For any $a, b in I$, $a + b in I$.
- Closure under multiplication by elements of $R$: For any $a in I$ and $r in R$, $r cdot a in I$ and $a cdot r in I$.
3. What is a subring?
Any non-empty part of a ring $R$ that can be worked on like $R$ is called a subring $S$ of $R$.
4. What is a quotient ring?
A quotient ring $R/I$ is a ring constructed from a ring $R$ and an ideal $I$, where elements of $R/I$ are equivalence classes of elements of $R$ modulo $I$.
5. What is a homomorphism?
A homomorphism between two rings $R$ and $S$ is a function $f: R to S$ that preserves the ring operations:
- $f(a + b) = f(a) + f(b)$
- $f(a cdot b) = f(a) cdot f(b)$
6. What is an isomorphism?
An isomorphism between two rings $R$ and $S$ is a bijective homomorphism, meaning that it is both a homomorphism and a bijection.
7. What is a polynomial ring?
A polynomial ring $R[x]$ over a ring $R$ is a ring consisting of polynomials in the variable $x$ with coefficients from $R$.
8. What is a module?
A module over a ring $R$ is an abelian group $M$ together with a scalar multiplication operation from $R$ to $M$, satisfying certain properties.
9. What is a Noetherian ring?
A Noetherian ring is a ring in which every ascending chain of ideals terminates.
10. What is Hilbert’s Basis Theorem?
Hilbert’s Basis Theorem states that every Noetherian polynomial ring is a Noetherian ring.
Additional Resources for Ring Theory
- Abstract Algebra by Dummit and Foote
- A First Course in Abstract Algebra by Fraleigh
- Ring Theory by Atiyah and Macdonald
- Math Stack Exchange
- InterviewPrep
By thoroughly understanding these ring theory interview questions and practicing your answers, you’ll be well-prepared to showcase your knowledge and impress your interviewers. Remember, confidence and a strong grasp of the concepts are key to success. So, study hard, practice your answers, and go into your interview with the confidence of a ring theory expert!
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As far as I know, there is no model theoretic proof of Faltings Theorem itself. Hrushovskis proof applies only to algebraic varieties over function fields and fails for varieties over number fields.
And yet, one of Abraham Robinson’s last works, a paper he wrote with Roquette and published after his death, Robinson, A ; Roquette, P. On the finiteness theorem of Siegel and Mahler concerning Diophantine equations, J. A proof of Siegel’s theorem on the finiteness of the number of integral points on curves of positive genus is given in Number Theory 7 (1975), 121–176.
“The complex numbers are easy to deal with, whereas for the integers it is much harder…”
You may have heard that the full first-order theory of complex numbers with only the ring operations can be decided. This is the same as the theory of algebraically closed fields of characteristic zero. e. “computable”), as was first proved by Tarski. Of course, you could make a computer program that would let you type in any first-order sentence in the language of rings and, after a certain amount of time, tell you if that sentence is true in the complex numbers or not.
The ring of integers, on the other hand, doesn’t say that; the first-order theory of the integers isn’t clear. This is a theorem of Alonzo Church, and is closely related to Goedels famous incompleteness theorem.
The negative answer to Hilbert’s Tenth Problem is a different matter. This doesn’t follow directly from Church’s Theorem; Davis, Putnam, Julia Robinson, and Matiyasevich proved it much later.
I think of these ideas as more “logical folklore” than model theory itself. Any good book for beginners on mathematical logic (e.g. g. Endertons A Mathematical Introduction to Logic) will have a lot to say about them.
A reasonably good “beginners” book on model theory is David Markers Model Theory: An Introduction. I’m not sure if he talks about the examples you’re thinking of, but he does talk about quantifier elimination and probably other tools that would be used in the proofs you want.
This book starts on a quite basic level and is packed with small applications of model theory to ring theory. Nothing as deep as Mordell-Lang, but it gives a good impression of how it is possible at all to derive statements about rings using logic.
That you have to deal with things like this is shown in a good way: it is ridiculously easy to prove (Thm 1 (See page 13 of the source above) that any two algebraically closed fields with the same property satisfy the same first-order ring claims One is amazed at first and tries to show something interesting about all char 0 fields by showing it for complex numbers. Then it is said: “It is hard to say something interesting in the first order language of rings; all you can really talk about are polynomial equations.” In these types of logic applications, the most important thing is often to come up with a clever way to say what you want to say in a language that is easy for everyone to understand (e.g. g. you have to find axioms whose resulting theory then satisfies quantifier elimination).
Hrushovskis proof, as it is presented in Bouscarens book, is very involving. It requires a lot of background, technical ingredients, and, lets say, a certain dose of faith. It is not a one-liner at all. Id rather recommend reading a more straight forward approach to the ch.0 case (using jet spaces of differential fields) given by A. Pillay and M. Ziegler and written out very carefully by Paul Baginski in his undergrad thesis.
I’m not sure about “short,” but I think Ehud Hrushovski gave the model-theoretic proof for Mordell’s Conjecture (Fatling’s Theorem).
(EDIT: and just to clarify, Im quite sure the proof is a good deal more complex than elimination of quantifiers per se. Also, Terry Tao is currently holding a reading seminar aimed at understanding another, more recent, paper of Hrushovski).
You might find this book helpful, if you want to get into the details. Disclaimer: I found it while googling for something else and figured Id mention it. I havent read it myself.
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Ideals in Ring Theory (Abstract Algebra)
FAQ
What are the fundamentals of ring theory?
What are the rules for the ring theory?
What are examples of associates in ring theory?
How do you prove ideal in ring theory?
What is ring theory in mathematics?
The ring theory in Mathematics is an important topic in the area of abstract algebra where we study sets equipped with two operations addition (+) and multiplication (⋅). In this article, we will study rings in abstract algebra along with its definition, examples, properties and solved problems. Let R be a non-empty set.
What is the interview process like at ring?
I interviewed at Ring in Sep 2022 It is mainly a technical interview. During the interview process, I mainly do code review and discuss program logic. The second period is mainly to ask questions to the interviewer and ask about company matters. An coding question about writing an class for phone book. I applied through a recruiter.
Why are polynomial rings so ubiquitous in ring theory?
An analogous statement is true for maps out of Z[x, y], etc. The last part of the proof suggests why polynomial rings are so ubiquitous in ring theory: polynomials are built entirely out of addition and multiplication, and homomorphisms pass through” addition and multiplication. So it is very easy to de ne homomorphisms on polynomials.
Do rings have nontrivial idempotents?
Products of rings also have nontrivial idempotents, which is a comparatively rare phenomenon. For instance, Z has no nontrivial idempotents, whereas Z Z has the idempotents (1, 0) and (0, 1).
