19.1. Discrete Mathematics

So, let us begin with the definition of a ring. So, we denote our ring by this notation ℝ and it is an algebraic structure. So, namely a set, set of values, it could be a finite set, it could be infinite set, so, it is a set and there are 2 operators plus (+) and dot (∙) which are defined over the elements of this set. I stress that these are not integer plus and integer dot they are just some abstract operations.

A ring is a foundational concept in abstract algebra, essentially consisting of a set of elements combined with two operations, typically addition and multiplication. The key here is that these operations are abstract and do not necessarily correspond to traditional arithmetic. The operations must be defined such that they adhere to specific rules, or axioms, that we will discuss further.

Think of a ring as a musical ensemble. Each musician (element of the set) has their own instrument (operation), but together they create a symphony (the algebraic structure). The way they interact (the abstract operations) doesn't have to reflect how musicians play in a typical orchestra setting.

But for the sake of notation we are using this plus notation and dot notation. So, we will say that this set ℝ along with these 2 operations plus and dot will be called a ring if all the following ring axioms are satisfied. Axiom number 1: we need the set ℝ to satisfy the properties of an abelian group with respect to your + operation.

A ring is defined by three axioms related to the operations that must hold true for the set elements. The first axiom requires the adherence to the properties of an abelian group for addition: closure (the result of adding any two elements is still in the set), associativity (the way in which elements are grouped doesn’t change the result), identity (there exists an element that, when added to any other, doesn’t change it), inverse (each element has an opposite that, when added, results in the identity), and commutativity (the order of addition doesn’t matter).

Imagine a group project. The team must work together (closure) without anyone dropping out (associativity), each member must contribute equally (identity), and if someone leaves, another must step in (inverse), plus it doesn't matter who presents first (commutativity).

The second axiom that set ℝ should satisfy is the following: we require that the dot operation should satisfy the closure property namely, you take any pair of elements (a, b) from your set ℝ and you perform the dot operation you should get back again an element from the same set ℝ.

The closure property indicates that when we apply the dot operation to any two elements of the ring, the result must also be an element of the same ring. This ensures that the system remains consistent within itself and does not produce elements outside the defined set.

Consider a factory producing toys. When the factory combines parts (elements), the result should always be a completed toy (the output must also be part of the toy set), ensuring that all products are usable and relevant to their purpose.

We require the dot operation to be associative. That means, it does not matter in what order you perform the dot operation on 3 elements, you should get back the same answer. And we demand the presence of an identity element with respect to the dot operation.

The associativity of the dot operation states that the grouping of elements does not affect the outcome of the operation. Additionally, there must be an identity element such that when multiplied with any element of the ring, the result is the element itself. This is crucial, as it provides a baseline or standard for multiplication in the ring.

Think about a group of friends deciding what movie to watch. It doesn't matter whether they first decide on the genre or the actors (associativity)—the final choice remains the same. Also, having a go-to classic movie (identity element) means that no matter what movies are discussed, they can always return to it if none seem appealing.

The third property or the so-called third ring axiom that needs to be satisfied is that your dot operation should be distributive over the plus.

Distributive property states that for any three elements a, b, and c in the ring, the operation of multiplication distributes over addition. This means that when you multiply a by the sum of b and c, it is the same as multiplying a by b and then a by c, then adding those results together. This axiom is critical in ensuring the interaction between the operations is consistent.

Think of it like sharing pizza. If you order one large pizza (a) and split it into two groups (b and c), each friend can get their share. No matter how you stack up the slices for each group, the total amount of pizza eaten remains the same.

Let us see some examples for ring. So, recall our set ℤ is the set of integers 0 to N - 1 and suppose I take 2 operations here: plus operation is the addition modulo N (+) and my multiplication operation is multiplication modulo N (∙)...

The example illustrates the concept of rings using the set of integers defined under modulo operations. When performing addition and multiplication under these operations, the set behaves according to the ring axioms, demonstrating closure, associativity, the presence of identity elements, and the distributive property.

Imagine a classroom where students can only pass on a piece of candy to another student without exceeding a certain number. When they add or multiply candies, they must still adhere to the classroom rules (modulo operations), but they follow all the rules effectively to ensure each student can still enjoy candy.

So, the next thing that we want to discuss is the invertible elements of a ring. So, imagine you are given a ring and if you see closely the ring axioms, it turns out that it is not the case that every element should have a multiplicative inverse...

Within a ring, not all elements have to possess a multiplicative inverse (an element which, when multiplied with the original element, yields the identity element of one). The subset of elements that do possess this property is termed U(ℝ), or the group of units, illustrating an important distinction within ring theory.

Consider a toolbox, where not all tools can work as replacements for each other (not all have an inverse). But some, like a screwdriver or wrench can be paired back with their proper counterparts offering functional versatility (invertibility).

So, we have to show the closure property. To prove this theorem that the collection of all invertible elements of an abstract ring constitutes a subgroup with respect to the dot operation...

To demonstrate that the invertible elements of a ring also form a subgroup under the dot operation, we must prove that the product of any two invertible elements remains invertible. The closure property, when proven, shows that the multiplicative nature of these elements is consistent and well-defined within the ring.

Think of a club of members. If every pair of members works together effectively (closure) to reach an end goal, they ensure that any projects they work on remain successful and yield results (they remain functional and valid within their community).

So now what we are going to do is we are going to extend our definition of ring to another interesting algebraic structure which we called as a field. So a field is an algebraic structure it is a set of values and there are 2 operations...

A field builds upon the concept of a ring but with stricter criteria. In a field, not only must addition and multiplication be defined and follow the ring axioms, but also every non-zero element must be invertible, meaning divisions and more complex algebraic operations can be performed naturally.

Consider a library system where every book (element) has a reference (inverse) in other systems. If someone can retrieve and reference any book without restrictions (field), the library operates efficiently, allowing for various interlibrary relational benefits.

The field axioms are as follows: the axiom number 1 is that the set F along with the plus operation should constitute an abelian group. The second property that we demand here is the following if I exclude the additive inverse and see...

The axioms of a field include forming an abelian group under addition; excluding the additive identity, the remaining elements form another abelian group under multiplication; and multiplication must be distributive over addition.

Imagine a dinner party where everyone shares their favorite dish (addition forms a group), and if you exclude the people who don’t participate (the additive identity), you would still have a full table enjoying their meals (the rest forming another group).

Now the next thing that we want to discuss here is the polynomials over rings, which is a very important concept used in computer science. So, imagine you are given a ring with some abstract plus an abstract dot operation...

Polynomials can be defined in a similar way as we defined them over integers, but here the coefficients come from a ring, and the operations used are those defined by the ring. This allows for the extension of polynomial algebra into more abstract realms while maintaining structure.

Think of creating a recipe where you're allowed to mix different ingredients (coefficients from the ring). Each recipe (polynomial) can have varied complexities depending on the available ingredients, adding to the variety yet keeping consistency with base cooking rules (operations from the ring).

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