Editing Maths (section)

==='''Ring'''===
Similar to a field but multiplication doesn't require an inverse.

'''ℤ, ℚ, ℝ, ℂ, ℤₘ''' are all '''unital commutative rings.'''

'''𝕄₂(ℝ)''' the set of all '''2×2 real matrices''' is a non-commutative '''ring''' as matrix multiplication is not commutative.

* '''+''' must be '''abelian'''.
* '''×''' must be '''closed''' and '''associative'''.
* [https://www.youtube.com/watch?v=syHBApgJhsA Bill Shillito - Rings and Fields]

* '''(R, +, ∗)'''
** '''(R, +)''' is an '''abelian group''' - '''addition'''
*** '''Closed'''
*** '''Associative'''
*** Identity - '''Additive identity'''
*** Inverse - '''Additive inverse −a'''
*** '''Commutative'''
** '''(R, ∗)'''
*** Closed under '''×'''
*** Operation '''×''' must be '''associative'''.
** '''+''' and '''×''' must be linked by the '''distributive property'''. Multiplication distributes over addition.
*** a×(b+c) = a×b+a×c
*** (a+b)×c = a×c+b×c
**** 0×a = a×0 = 0 - Zero times anything is zero
**** a×−b = −a×b = −(a×b) - A positive times a negative is a negative.
**** −a×−b = a×b - A negative times a negative is a positive.

* '''×''' '''might''' not be '''commutative'''.  If it is then it is a '''commutative ring''' (not called abelian which is only for groups).
* If the ring has a '''multiplicative identity''' then it is a '''unital ring'''. For '''ℤ, ℚ, ℝ, ℂ, ℤₘ''' the '''multiplicative identity''' is 1. For '''𝕄ₘ(ℝ)''' it is the '''identity matrix'''.
* An element of a unital ring that has a multiplicative inverse is called a '''unit'''. Not every element is necessarily a unit. In ℤ only 1 and −1.
* If '''every element other than 0 is a unit'''. It is a '''division ring'''. ℚ is a division ring.
* A '''zero divisor''' of a ring is a nonzero element that can be multiplied by some other nonzero element to produce 0. In ℤ₆, 2, 3 and 4 are zero divisors. 2×3=0
* An element '''can not''' be a '''unit''' and a '''zero divisor'''.
* An '''integral domain''' is a '''commutative, unital ring''' with '''no zero divisors'''. '''ℤ, ℚ, ℝ, ℂ''' are '''integral domains'''.
** ℤₘ is not necessarily an integral domain. '''ℤₚ''' is an integral domain '''if p is a prime number'''.
** In an integral domain you can '''cancel factors''' from both sides of an equation. '''<strike>2</strike>×x=<strike>2</strike>×3'''

====Polynomial Ring====
A '''monomial''' is the product of a number and a non-negative integer power of a variable. eg: 2x³

A '''polynomial''' is a finite sum of monomials. a₀+a₁x+a₂x²+a₃x³+…+aₙxⁿ

n is the degree of the polynomial. It's highest power.

Some polynomial rings: ℤ[𝓍], ℚ[𝓍], ℝ[𝓍], ℂ[𝓍], ℤₘ[𝓍]

In '''ℝ[x]''' we can take congruents to the modulus '''𝓍²+1'''. Two polynomials are congruent if their difference is divisible by '''𝓍²+1'''. This makes the polynomial ring behave like complex numbers. When using a ''''prime' polynomial''' as modulus, the '''polynomial ring''' is a '''field'''.

'''𝓍³+𝓍²-2𝓍+3 ≡ -3𝓍+2   (mod 𝓍²+1)'''

Mentioned in a concrete guide to modern mathematics, page 90.