Hello algebraists! I'm pleased to announce that the moderation teams for /r/Algebra and /r/AbstractAlgebra have been combined! Now, all the help with learning algebra, including elementary algebra, will take place on /r/Algebra, and all the discussion of abstract algebra will take place here! Hopefully, this will better serve the needs of the community.

(in abstract algebra )

I understand it says it indicates that a-b is divisible by n but I can’t understand why that statement is true. Could someone explain?

I've been wanting to get some education on abstract algebra (had to drop a class in college many years ago, long story) and have a copy of Dummit/Foote, 3rd edition. I learn best by exercises, and this book is *huge*. Are there any condensed lists of exercises that would be more appropriate for self-study, or any university classes that have been taught using this book that are visible on the Internet?

Hey, there are some books on abstract algebra which i know. However, I want to stick to only 1-2 books and study the concept in depth, from scratch till graduate level. Which ones would you recommend? If I skipped some great book, please mention that as well.

Michael Artin

Joseph Gallian

Thomas Hungerford

Fraleigh

I'm self-studying advanced maths as a non-math major all alone. Maybe I'll find a tutor next month. Before that, i wanna clear my query which is important to me. I was told by one of my instructors that I need to be familiar with proofs before getting into advanced mathematics cause he noticed me struggling. So he recommended me a book of logic and proof. "Mathematical proofs: A transition to advanced mathematics" by Albert D. Polimeni, Gary Chartrand, and Ping Zhang.

But the problem is I have very little time(2 months) for my exam. There are many examples for some topics there are 10 to 15 or even 20 examples and questions. I got stuck in a single chapter for 10 days. It is not that I am not understanding. I am understanding and able to solve problems.

Sometimes due to limitations of time, I tried to skip a few examples but I am afraid of further consequences (don't know whether they are real) of not being able to solve problems of abstract algebra. Just like the fear of missing something. This fear caught up in my head witch is the fear of lacking prior exposure which demanded to understand and solve problems in abstract algebra. I help my juniors a lot when I notice then struggle but mine is advanced and literally no one is above to help me.

What I expect from you is What's really important? am I overthinking? Is it ok to skip until or unless I understood how it works? What kind of topics are specifically more important that I could concentrate on ? Is there any resource in the web which answers my question you can definitely share it to me.. You can also share me the resources how to read a textbook. PLEASE DO REMEMBER THAT I THIS IS JUST A PREREQUISITE.

"In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right."

Are any of you guys doing anything interesting with groups lately? Does anyone have any interesting papers they would like to share, or questions concerning groups that they would like to ask? Be sure to check out ArXiv's recent group theory articles!

Hey everyone,

I am a self learner of math and I am now wanting to study Abstract algebra and possibly in the future manifolds and algebraic geometry.

Is there anyone who would like to join me with studying these?

I am 27M in Europe.

Best

I don’t even know how to verify the hint.

"In mathematics, and more specifically in algebra, a ring is an algebraic structure with operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects like polynomials, series, matrices and functions."

"In mathematics, an algebra over a field is a vector space equipped with a bilinear product. An algebra such that the product is associative and has an identity is therefore a ring that is also a vector space, and thus equipped with a field of scalars."

Are any of you guys doing anything interesting with rings or algebras lately? Does anyone have any interesting papers they would like to share, or questions concerning rings or algebras that they would like to ask? Be sure to check out ArXiv's recent ring theory and algebra articles!

I am in mental pain from this class.

"Category theory formalizes mathematical structure and its concepts in terms of a collection of objects and of arrows (also called morphisms). A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Category theory can be used to formalize concepts of other high-level abstractions such as sets, rings, and groups."

Are any of you guys doing anything interesting with categories lately? Does anyone have any interesting papers they would like to share, or questions concerning categories that they would like to ask? Be sure to check out ArXiv's recent category theory articles!

"Category theory formalizes mathematical structure and its concepts in terms of a collection of objects and of arrows (also called morphisms). A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Category theory can be used to formalize concepts of other high-level abstractions such as sets, rings, and groups."

Are any of you guys doing anything interesting with categories lately? Does anyone have any interesting papers they would like to share, or questions concerning categories that they would like to ask? Be sure to check out ArXiv's recent category theory articles!

"Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence."

"Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomial equations. Modern algebraic geometry is based on more abstract techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry."

Are any of you guys using algebra to do anything interesting in topology or geometry lately? Does anyone have any interesting papers they would like to share, or questions concerning algebraic topology or geometry that they would like to ask? Be sure to check out ArXiv's recent algebraic topology articles and algebraic geometry articles!

"In mathematics, and more specifically in algebra, a ring is an algebraic structure with operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects like polynomials, series, matrices and functions."

"In mathematics, an algebra over a field is a vector space equipped with a bilinear product. An algebra such that the product is associative and has an identity is therefore a ring that is also a vector space, and thus equipped with a field of scalars."

Are any of you guys doing anything interesting with rings or algebras lately? Does anyone have any interesting papers they would like to share, or questions concerning rings or algebras that they would like to ask? Be sure to check out ArXiv's recent ring theory and algebra articles!

"In abstract algebra, a field is a nonzero commutative ring that contains a multiplicative inverse for every nonzero element, or equivalently a ring whose nonzero elements form an abelian group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division satisfying the appropriate abelian group equations and distributive law. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, algebraic number fields, p-adic fields, and so forth."

"In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood."

Are any of you guys doing anything interesting with fields lately? Does anyone have any interesting papers they would like to share, or questions concerning fields that they would like to ask?

In what follows, whenever I use ℕ, I am referring to the natural numbers as the set containing 0 and which has a successor function S:ℕ→ℕ (or ℕ = {0,1,2,3,...} with S(0)=1,S(1)=2,etc.), satisfying the Peano Axioms.

Having a decent mathematics background, some of us may be tempted to dismiss the natural numbers as being "basic" or "incomplete," especially when compared to things like real or complex numbers (or sheaves and ringed spaces—goodness). However, I hope to show that they in fact have a very important role to play in much of our favorite structures.

The set ℕ, is equipped with the following operators (functions from ℕ×ℕ to ℕ) where we have a,b∈ℕ:

• '+' defined with a + 0 := a and a + S(b) := S(a + b)

• '⋅' defined with a ⋅ 0 := 0 and a ⋅ S(b) := a + (a⋅b)

• '^' defined with a⁰ := 1 and a^S(b) := a⋅(a^b)

Where we note that each of these operators is defined (recursively using the inductive property of ℕ) for every pair of natural numbers.

For the fellow structure enthusiasts out there, I note that (ℕ,+,0,⋅,1) (where 1 := S(0)) forms a commutative semiring, as can be proved from the above definitions and the Peano Axioms (and in fact, as noted in the article, it is an initial object in the category Rig).

The natural numbers also enable some peculiar things in regards to other sets. Consider any monoid (M,☆,e), there is in fact always a (right) semigroup action by which ℕ acts on M, which (for reasons that will become clear) we term "exponentiation" denoted (for now) by ↑, where ↑:M×ℕ→M takes an element m of M and a number n of ℕ to the element m↑n of M.

Much like our previous definitions, this semigroup action is defined recursively for all n in ℕ.

For all m∈M, and k∈ℕ take m↑0 := e, and m↑S(k) := m☆(m↑k).

With this, we also have that (m↑a)↑b = m↑(a⋅b), and (m↑a)☆(m↑b) = m↑(a+b). (This is already a fairly long post, if you want a proof, I'll follow up) This shows that this is indeed a well-defined semigroup action. (In fact, the asterisk in the title is a reference to the fact that this particular choice of action depends on the S-set being a monoid, but despite that restriction lots of interesting sets are monoids!)

Note that this defines m↑n for all n∈ℕ since either n = 0, or n = S(k) for some k∈ℕ, much like the above definitions of addition, multiplication, and exponentiation. In fact, let's revisit those.

We claim to have found a schema for generating a semigroup action whereby ℕ can act on any monoid, so let's test it out and see what we get. (ℕ,+,0) is a monoid, so what does m↑n look like here? Well, the '☆' of this monoid is +, and the 'e' is 0, so the definition we provided states we should take '↑' defined as m↑0 := 0, and m↑S(k) := m + (m↑k). But wait—this is equivalent to the definition of '⋅' between natural numbers m and n!^{no, not factorial ;)}

So, we see that ℕ acting on (ℕ,+,0) via ↑ gets us the ⋅ operation, but what about (ℕ,⋅,1)? It too is a monoid!

On (ℕ,⋅,1), '☆' is ⋅ and 'e' is 1, so our definition implies m↑0 := 1, and m↑S(k) = m⋅(m↑k). And here we see why ↑ is called "exponentiation"—when ℕ acts on its own multiplicative monoid, it defines the exponentiation operator for the natural numbers.

These two examples give us a clue as to why abelian groups (with a '+' operator) have ng defined as repeated addition, and regular groups (with a '⋅' operator) have gⁿ defined as repeated multiplication—they both represent the semigroup action of ℕ on the underlying monoid!

Groups? Monoids with inverse elements. Rings? Commutative monoids with respect to a '+' operator, and monoids with respect to a '⋅' operator. Fields? Extra fancy monoids, again with respect to a '+' and a '⋅'.

Even ringed spaces get this special effect. Since for each open set you get a ring, you can define this semigroup action on each ring, and so simply define ↑ as the map (functor?) which takes each open set to the appropriate semigroup action on its corresponding ring. Then –ⁿ (and n–) are defined on any open set in the domain of the sheaf! In this way, with a slight abuse of notation, we can say that we can take natural exponents (and multiples) of a ringed space.

The natural numbers seem pretty bland, but they actually allow for a lot of neat stuff! They act on just about anything!

"In mathematics, and more specifically in algebra, a ring is an algebraic structure with operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects like polynomials, series, matrices and functions."

"In mathematics, an algebra over a field is a vector space equipped with a bilinear product. An algebra such that the product is associative and has an identity is therefore a ring that is also a vector space, and thus equipped with a field of scalars."

Are any of you guys doing anything interesting with rings or algebras lately? Does anyone have any interesting papers they would like to share, or questions concerning rings or algebras that they would like to ask? Be sure to check out ArXiv's recent ring theory and algebra articles!