Application of ring theory in computer science

By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. This is ONE thing about my undergraduate studies in computer science that I haven't been able to 'link' in my real life academic and professional.

Almost everything I studied I've observed be applied directly or indirectly or has given me Aha! Groups, Rings and Fields have always eluded me. Are they just theoretical concepts without practical applications? I hope not. So what are their applications, especially in the field of computer science.

Groups and fields, primarily finite ones, are used extensively in coding theory. Many of the results in number theory that give rise to important encryption systems e. If you include applications outside of computer science it would really be hard to exaggerate on the importance of group theory. Groups are literally everywhere. The theory of group representations for instance is useful in chemistry particularly in crystallography. The reason for the importance of groups is that they model symmetry and for fields, at least for coding theory and cryptography, is that they codify very intricate combinatorics.

So, in computer science, whenever you watch a video online, make a phone-call, purchase something over the internet, compress a file, send an email, or communicate with the Mars Rover lots of groups and fields are being used behind the scenes. This is almost certainly not true if the counter could only be set to 1, 2, 3, or 4.

Saying my bit and it's too long to fit into a comment. I apologize in advance about being a bit chatty. Well, this is a soft question, so the answer is gonna be soft as well. Several posters have emphasized some technological applications of abstract algebra.

From the point of view of algebraists these are fine answers surprisingly generously upvoted actually - may be the hearts of practitioners of abstract algebra warm up to these.

application of ring theory in computer science

You will get different answers, if you ask a different group of people. Some might refer to how algebraic structures give the playground to problems central to study of complexity classes of algorithms.

I dunno? I once chatted with a professor in computer science. I suggested that may be I should supplement our linear algebra lecture notes with a chapter on how orthogonal coordinate transformations rotations and such are applied in 3D-graphics. I had found out that homework problems related to this theme motivated some of my students.In physics the relation of groups with symmetries means that group theory plays a huge role in the formulation of physics.

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Fundamental in modern physics is the representation theory of Lie groups. Lie groups like the Poincare group, SU nO n etc all play fundamental roles in physics. In chemistry group theory is used to describe symmetries of crystal and molecular structures. This is then important in understanding the physical and spectroscopic properties of materials, for example. Probably, group theory is the most powerful branch of mathematics when it comes to quantum chemistry, spectroscopy and condensed matter physics.

Real numbers form an abelian group under addition and non-zero real numbers form an abelian group under standard multiplication.

We have a commutative ring, in fact we have a field. These are the only thing that springs to mind in "everyday life". For example, the fact that the real numbers form an abelian group under addition is used when working out change when you buy something. That said, it maybe a bit of an overkill. Group actions can be performed on decks or hands of cards - so that could be considered an application of group theory, even though you wouldn't really think of it like that. An interval is the distance from one note to the another — e.

C—C is an interval of a semitone, C—D is a whole-tone interval, C—D is an interval of a minor third, etc. Note that the starting note can be any note, so F—F is also a semitone interval. The unison interval is the interval from one to itself e. All intervals that are whole octaves can be identified with the unison interval. It follows that the set of all intervals under this addition operation forms a group, the cyclic group of order The identity element is the unision interval, and the group is generated by four intervals: semitone C—Cperfect fourth C—Fperfect fifth C—Gand major seventh C—B.

This cyclic group of order 12 is the basis on the theory of the circle of fifths. It also explains why there are only two whole-tone scales — namely, because the subgroup generated by the whole-tone interval C—D is a subgroup of order 6 and so has index 2.

Very true. I use the symmetry operations on a daily basis in analyzing UV spectroscopy [looking for specific electronic transitions]. The concepts were difficult to get a handle on at first, but well worth it. Maybe one day DFT will advance to the level to were I don't have to consciously remember symmetry operations and point groups. That would be nice. All the above are true.

But for an everyday mundane application, balancing your ckeckbook uses only the abelian group operations of the integers. According to the maths, hydrated copper sulphate isn't blue; Cobalt chloride is white and certainly doesn't change colour from blue to pink in response to humidity changes; and rubies are colourless. Another good example are permanganates, which are purple due to a ligand to metal charge transfer band or maybe metal to ligandA ring topology is a network configuration where device connections create a circular data path.

Each networked device is connected to two others, like points on a circle. Together, devices in a ring topology are referred to as a ring network. In a ring network, packets of data travel from one device to the next until they reach their destination. Most ring topologies allow packets to travel only in one direction, called a unidirectional ring network. Others permit data to move in either direction, called bidirectional.

The major disadvantage of a ring topology is that if any individual connection in the ring is broken, the entire network is affected. Depending on the network card used in each computer of the ring topology, a coaxial cable or an RJ network cable is used to connect computers together. In the past, the ring topology was most commonly used in schools, offices, and smaller buildings where networks were smaller. However, today, the ring topology is seldom used, having been switched to another type of network topology for improved performance, stability, or support.

Network termsRing statusToken RingTopology. Home Dictionary R - Definitions. Ring topology history. Advantages of a ring topology.

Disadvantages of a ring topology. Related pages. Related pages Network and network card help and support. Was this page useful?Algebraic number theory. Noncommutative algebraic geometry. In abstract algebraa semiring is an algebraic structure similar to a ringbut without the requirement that each element must have an additive inverse.

The term rig is also used occasionally [1] —this originated as a joke, suggesting that rigs are ri n gs without n egative elements, similar to using rng to mean a r i ng without a multiplicative i dentity. Tropical semirings are an active area of research, linking algebraic varieties with piecewise linear structures.

Compared to a ringa semiring omits the requirement for inverses under addition; that is, it requires only a commutative monoidnot a commutative group. In a ring, the additive inverse requirement implies the existence of a multiplicative zero, so here it must be specified explicitly. If a semiring's multiplication is commutativethen it is called a commutative semiring. There are some authors who prefer to leave out the requirement that a semiring have a 0 or 1.

This makes the analogy between ring and semiring on the one hand and group and semigroup on the other hand work more smoothly. These authors often use rig for the concept defined here.

application of ring theory in computer science

Much of the theory of rings continues to make sense when applied to arbitrary semirings [ citation needed ]. In particular, one can generalise the theory of associative algebras over commutative rings directly to a theory of algebras over commutative semirings. Then a ring is simply an algebra over the commutative semiring Z of integers. These dynamic programming algorithms rely on the distributive property of their associated semirings to compute quantities over a large possibly exponential number of terms more efficiently than enumerating each of them.

By definition, any ring is also a semiring. A motivating example of a semiring is the set of natural numbers N including zero under ordinary addition and multiplication. Likewise, the non-negative rational numbers and the non-negative real numbers form semirings.

application of ring theory in computer science

All these semirings are commutative. A semiring of sets [14] is a non-empty collection S of sets such that. Such semirings are used in measure theory. An examples of a complete semiring is the power set of a monoid under union. The matrix semiring over a complete semiring is complete.

A continuous semiring is similarly defined as one for which the addition monoid is a continuous monoid. That is, partially ordered with the least upper bound propertyand for which addition and multiplication respect order and suprema. Any continuous semiring is complete: [18] this may be taken as part of the definition.

A Kleene algebra is a star semiring with idempotent addition. They are important in the theory of formal languages and regular expressions. In a complete star semiringthe star operator behaves more like the usual Kleene star : for a complete semiring we use the infinitary sum operator to give the usual definition of the Kleene star: [16].

A Conway semiring is a star semiring satisfying the sum-star and product-star equations: [7] [25]. Every complete star semiring is also a Conway semiring, [26] but the converse does not hold. An iteration semiring is a Conway semiring satisfying the Conway group axioms, [7] associated by John Conway to groups in star-semirings. A generalization of semirings does not require the existence of a multiplicative identity, so that multiplication is a semigroup rather than a monoid.

Such structures are called hemirings [31] or pre-semirings. Yet a further generalization are near-semirings : in addition to not requiring a neutral element for product, or right-distributivity or left-distributivitythey do not require addition to be commutative.

Just as cardinal numbers form a class semiring, so do ordinal numbers form a near-ringwhen the standard ordinal addition and multiplication are taken into account.Authors: BiniGilberto, FlaminiFlaminio. Foreword by Dieter Jungnickel Finite Commutative Rings and their Applications answers a need for an introductory reference in finite commutative ring theory as applied to information and communication theory.

application of ring theory in computer science

This book will be of interest to both professional and academic researchers in the fields of communication and coding theory. The book is a concrete and self-contained introduction to finite commutative local rings, focusing in particular on Galois and Quasi-Galois rings. The reader is provided with an active and concrete approach to the study of the purely algebraic structure and properties of finite commutative rings in particular, Galois rings as well as to their applications to coding theory.

Finite Commutative Rings and their Applications is the first to address both theoretical and practical aspects of finite ring theory. The authors provide a practical approach to finite rings through explanatory examples, thereby avoiding an abstract presentation of the subject.

The section on Quasi-Galois rings presents new and unpublished results as well. The authors then introduce some applications of finite rings, in particular Galois rings, to coding theory, using a solid algebraic and geometric theoretical background.

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About this book Foreword by Dieter Jungnickel Finite Commutative Rings and their Applications answers a need for an introductory reference in finite commutative ring theory as applied to information and communication theory.

Show all. Finite Commutative Rings. Regular Polynomials Pages Bini, Gilberto et al. Show next xx. Read this book on SpringerLink. Recommended for you. PAGE 1.A huge significant switch of odds at the last moment may mean that something shady is going on, mostly in very low tier games, or a sudden announcement, E. Bookies will put up their odds based on their own expectations, as the bets start coming in they'll tweak the odds to ensure it's still profitable for them.

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