19.2. Rings, Fields and Polynomials

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. 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.

A ring is defined as a set equipped with two operations, typically referred to as addition (denoted by '+') and multiplication (denoted by '∙'). These operations are not limited to the usual addition and multiplication of integers; rather, they are abstract operations defined over the elements of the set. For a set to qualify as a ring, it must meet certain axioms clearly outlined in later sections.

Think of a ring like a toolbox. The set is your toolbox, and the operations are the tools (like a hammer for addition and a screwdriver for multiplication). If you can use your tools effectively according to certain rules (axioms), then you have a well-functioning toolbox or ring.

Axiom number 1: we need the set ℝ to satisfy the properties of an abelian group with respect to your + operation. We need closure property, we need plus operation to be associative, we require the presence of some special identity element which we denote by this ‘0’ such that if you perform the plus operation with element a you should get back the element a for every element a from the set ℝ. We need the presence of additive inverse and we need the operation plus to be commutative.

The first axiom requires that the set ℝ with the addition operation (+) behaves like an abelian group. This means that: 1) there is closure under addition (adding two elements results in an element in the set), 2) the addition is associative (the grouping of additions does not affect the outcome), 3) there is an identity element (0) such that adding it to any element x returns x, 4) each element has an additive inverse (for every x, there is a -x such that x + (-x) gives 0), and 5) addition is commutative (x + y = y + x).

Imagine a popular card game where players can only add points to their score. The rules dictate that if you draw certain cards, you can only add to your score. The first rule of being able to add your score without exceeding the maximum score reflects the closure property, just like in a ring. Every player can also exchange points (add inverses), ensuring fairness in the game.

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. We require the dot operation to be associative. And we demand the presence of an identity element with respect to the dot operation.

The second axiom focuses on the dot operation (multiplication). It states that for any two elements in the set ℝ, the result of their multiplication should also be in ℝ (closure). Additionally, the multiplication operation must be associative, meaning the order in which you multiply does not affect the result. There must exist an identity element (denoted 1) such that when you multiply it with any element a, you get back a (a∙1 = a).

Consider baking in a kitchen. If you add ingredients (dot operation) and you only use things available in your pantry (closure), it's as if all mixes yield a dish from your pantry. If you pour milk and then add sugar, it doesn't matter if you first mix them together or add them separately (associativity), and the milk acts as the identity, maintaining the same base taste.

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. If you take any triplet of elements from the set ℝ called them as a, b, c, then it does not matter whether you first perform the plus operation on b and c.

The third axiom establishes the distributive property of multiplication over addition. This means if you multiply an element by the sum of two other elements, it should be the same as multiplying the element by each of the added elements first and then adding the results. Formally, this can be expressed as a∙(b + c) = a∙b + a∙c. This property must hold true in both directions (left and right).

Think of distributing candies among friends. If you have a bag of candies (the total) and you want to share some with your two friends (addition), you can either pour candies from the bag to each friend one by one or combine the candies meant for each friend first and then share from the total (distributive property). The total candies remain the same regardless of how you distribute.

Recall our set ℤ is the set of integers 0 to N - 1and N. Suppose I take 2 operations here: plus operation is the addition modulo N (+) and my multiplication operation is multiplication modulo N (∙).

An example of a ring can be the set of integers modulo N, denoted ℤ. For instance, if we choose N = 4, the elements are {0, 1, 2, 3}. Here, addition and multiplication are performed modulo 4 (e.g., 2 + 3 = 1 in modulo 4). This set satisfies all ring axioms, making it a valid ring.

Imagine a clock with numbers from 0 to 11. If it's 10 o'clock now, and you add 5 hours, you arrive at 3 o'clock due to wrapping around (modular addition). The number of hours represents elements in ℤ, and the way we add or multiply the hours represents the operations on a ring.

Now we want to discuss the invertible elements of a ring. It is not the case that every element should have a multiplicative inverse. The presence of identity element 1 is required, but not every element has a multiplicative inverse.

Not all elements in a ring are invertible; meaning not every element will have a multiplicative inverse. An element 'x' in a ring has a multiplicative inverse if there exists another element 'y' such that x∙y = 1 (the identity element). For instance, in the ring ℤ_4, the element 2 does not have an inverse because no element multiplied by 2 gives 1 (mod 4).

Picture a school club where each member has a unique badge number. Only some members can swap badges (invert) with others to create pairs that form complete sets. For example, if a member number 2 cannot find a partner to form a complete pair from the club's set of badges, they are like an element without an inverse—existing but unable to fully participate in the exchange.

Now we are going to define a special set which I call as U(ℝ) which is basically the collection of all invertible elements with respect to the multiplication operation.

The set U(ℝ) is defined as the collection of all elements in a ring ℝ that are invertible under multiplication. An element 'x' belongs to U(ℝ) if there is some element 'u' such that x∙u = 1 (the multiplicative identity). These invertible elements play a critical role in the structure of the ring.

In the context of a sports league, think of players with unique abilities that allow them to exchange roles (invert) with each other during a game. Only certain players (invertible elements) possess the versatility to switch positions dynamically, maintaining the team's balance and strategy during the game.

Now we are going to extend our definition of ring to another interesting algebraic structure which we called as a field. A field is a set of values and there are 2 operations; 2 abstract operations plus and dot, which are defined over the elements of this set F.

A field is a specific type of ring with additional properties. All the elements of the field, except for the additive identity (0), have multiplicative inverses. Thus, in a field, both addition and multiplication (excluding zero) must behave like abelian groups. The requirement for invertibility adds a new layer to the definitions we have outlined for rings.

If you think of baking as a ring activity, where you mix flour and sugar, a field can be likened to a fully equipped bakery where every ingredient can be duplicated or swapped inexpensively. Here, each ingredient's flexibility (invertibility) opens up possibilities for endless delightful recipes, ensuring that even when something is used (like 0), the rest can still create delicious treats.

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