Mathematics - Discrete Mathematics - Rings

Introduction

Hey it's a me again @drifter1!

Today we continue with Mathematics, and more specifically the branch of "Discrete Mathematics", in order to get into Rings.

So, without further ado, let's get straight into it!

Rings

Rings are yet another algebraic structure, which is defined as a set together with two binary operations on that set. The binary operations have to satisfy properties similar to those of addition and multiplication of integers. It's thus common to note these binary operations as "+" and "." respectively. Rings are thus denoted as (R, +, .), where R is the set and "+", "." the binary operations on that set.

Ring Axioms

Rings satisfy the following three axioms:

(R, +) is an abelian group, which means that it satisfies:
- associativity
- commutativity
- (additive) identity
- (additive) invertibility
(R, .) is an monoid, which means that it satisfies:
- associativity
- (multiplicative) identity
Multiplication (.) is distributive over addition (+)

Distributivity is the following property:

Commutative Rings

Commutative rings are rings, which additionally satisfy commutativity under the operation of multiplication, basically:

Non-commutative Rings

Rings which don't satisfy commutativity under either of the two binary operations are known as non-commutative rings.

For example, in the case of any ring R and natural number n, the set of all square n x n matrices forms a ring under matrix addition and multiplication, which is non-commutative.

Semirings

Rings which don't satisfy additive invertibility are known as Semirings.

For example, the set of natural numbers together with addition and multiplication is a semiring, because not all elements are invertible under addition.

Subrings

A subset A of a ring (R, +, .) is a subring of R if (A, +) is a subgroup of (R, +) and A is closed under multiplication.

In that context, a set consisting only of 0 or {0} and of course R itself are directly also subrings of R.

Rng

In some definitions rings are defined without the requirement of multiplicative identity. Such structures are more commonly known as rng's.

RESOURCES:

References

Images

https://commons.wikimedia.org/wiki/File:Pullback_commutative_rings.svg

Mathematical equations used in this article, have been generated using quicklatex.

Block diagrams and other visualizations were made using draw.io.

Previous articles of the series

Introduction → Discrete Mathematics, Why Discrete Math, Series Outline
Sets → Set Theory, Sets (Representation, Common Notations, Cardinality, Types)
Set Operations → Venn Diagrams, Set Operations, Properties and Laws
Sets and Relations → Cartesian Product of Sets, Relation and Function Terminology (Domain, Co-Domain and Range, Types and Properties)
Relation Closures → Relation Closures (Reflexive, Symmetric, Transitive), Full-On Example
Equivalence Relations → Equivalence Relations (Properties, Equivalent Elements, Equivalence Classes, Partitions)
Partial Order Relations and Sets → Partial Order Relations, POSET (Elements, Max-Min, Upper-Lower Bounds), Hasse Diagrams, Total Order Relations, Lattices
Combinatorial Principles → Combinatorics, Basic Counting Principles (Additive, Multiplicative), Inclusion-Exclusion Principle (PIE)
Combinations and Permutations → Factorial, Binomial Coefficient, Combination and Permutation (with / out repetition)
Combinatorics Topics → Pigeonhole Principle, Pascal's Triangle and Binomial Theorem, Counting Derangements
Propositions and Connectives → Propositional Logic, Propositions, Connectives (∧, ∨, →, ↔ and ¬)
Implication and Equivalence Statements → Truth Tables, Implication, Equivalence, Propositional Algebra
Proof Strategies (part 1) → Proofs, Direct Proof, Proof by Contrapositive, Proof by Contradiction
Proof Strategies (part 2) → Proof by Cases, Proof by Counter-Example, Mathematical Induction
Sequences and Recurrence Relations → Sequences (Terms, Definition, Arithmetic, Geometric), Recurrence Relations
Probability → Probability Theory, Probability, Theorems, Example
Conditional Probability → Conditional Probability, Law of Total Probability, Bayes' Theorem, Full-On Example
Graphs → Graph Theory, Graphs (Vertices, Types, Handshake Lemma)
Graphs 2 → Graph Representation (Adjacency Matrix and Lists), Graph Types and Properties (Isomorphic, Subgraphs, Bipartite, Regular, Planar)
Paths and Circuits → Paths, Circuits, Euler, Hamilton
Trees → Trees (Rooted, General and Binary), Tree Traversal, Spanning Trees
Common Graph Problems → Shortest Path Problem, Graph Connectivity, Travelling Salesman Problem, Minimum Spanning Tree, Maximum Network Flow, Graph Coloring
Binary Operations → Binary Operations (n-ary, Table Representation), Properties
Groups → Groups (Properties, Theorems, Finite and Infinite, Abelian, Cyclic, Product, Homo-, Iso- and Auto-morphism)
Group-like Structures (part 1) → Subgroups, Semigroups, Monoids
Group-like Structures (part 2) → Magma, Quasigroup, Groupoid

Final words | Next up

And this is actually it for today's post!

Next time we will cover Fields...

See ya!

Keep on drifting!

Posted with STEMGeeks