Applications of Algebra: Groups, Rings, Ideals

[matness]

always abstract?

things are really very abstract in algebra as its name.But it would be nicer to learn some applications especially to physics
apperently there are applications about groups but i could not find anything related to rings ideals etc.
discussing the applications would make it more meaningfull at least for me

thanks in advance
n

 

[matt grime]

Read jon baez's this weeks finds. there are too many things to talk about here, and we have no idea what you do and do not understand. for instance does the phrase: a topological quantum field theory is esseentially a monoidal category with a functor to a category of riemann surfaces and cobordisms mean anything to you? i suspect not; it means little to me.

the group applications are an application of rings (the group ring) and lie algebras are used in physics, you look at the ring of operators on a hilbert space, rings occur everywhere because matrices are rings.

 

[tacman]

There are many practical applications of algebra, including in the fields of physics, computer science, and cryptography. In fact, groups, rings, and ideals are all important concepts in these fields.

Groups, which are sets with a binary operation that follows certain rules, have many applications in physics. For example, in quantum mechanics, groups are used to describe the symmetries of physical systems. This allows for a deeper understanding of the underlying principles governing these systems. Groups are also used in crystallography to describe the symmetries of crystals.

Rings, which are sets with two operations (addition and multiplication) that follow certain rules, have applications in computer science and cryptography. In computer science, rings are used to study abstract data types and algorithms. In cryptography, rings are used to create secure encryption algorithms.

Ideals, which are subsets of rings that behave similarly to normal numbers, also have applications in cryptography. Specifically, ideals are used in the construction of public-key cryptosystems, which are essential for secure communication over the internet.

So while algebra may seem abstract, its applications in various fields are very real and important. Understanding concepts like groups, rings, and ideals can lead to advancements and innovations in these fields, making algebra a valuable and relevant subject to study.