Number Theory Part 2: Ring Theory
In continuation with my previous blog, this blog will discuss the ring theory.
Ring
Discovering new ways to interact with abelian groups can lead us to unlock the potential of rings, which is an important stepping stone towards understanding the building blocks of modern cryptography. So, the ring is an abelian group under addition. A ring is a set equipped with two binary operations (usually referred to as addition, +, and multiplication, ×).
Mathematically, A ring is typically denoted {R, +, ×} where R denotes the set of objects, ’+’ the operator with respect to which R is an abelian group, the ’×’ the additional operator needed for R to form ring
Ring satisfies certain A1-A5 properties discussed with abelian groups, including the following properties:
M1. Closure under multiplication: R must be closed with respect to the additional operator ’×’.
M2. Associative under multiplication: R must exhibit associativity with respect to the additional operator ‘×’.
M3. Distributivity: The multiplication operation is distributive over the addition operation, which means for all elements a, b, and c in the ring,
a×(B+c)=a×b+a×c
(a+b)×c=a×c+b×cLet’s understand with examples:
- The set of polynomials with real coefficients under addition and multiplication: The set of polynomials with real coefficients is closed under addition and multiplication because the addition and multiplication of any two polynomials with real coefficients are also a polynomial with real coefficients. The operations of addition and multiplication are commutative because p(x) + q(x) = q(x) + p(x) and p(x) × q(x) = q(x) × p(x) for all polynomials p(x) and q(x) with real coefficients. The operations of addition and multiplication are associative because (p(x)+q(x))+r(x)=p(x)+(q(x)+r(x)) and (p(x) × q(x)) × r(x)=p(x) × (q(x) × r(x)) for all polynomials p(x), q(x), and r(x) with real coefficients. The additive identity element is the polynomial 0(x), and the multiplicative identity element is the polynomial 1(x). Every polynomial has an additive inverse, and every nonzero polynomial has a multiplicative inverse.
- The set of square matrices with real coefficients under addition and matrix multiplication: The set of matrices with real coefficients is closed under addition and matrix multiplication because the addition and multiplication of any two square matrices with real coefficients is also a matrix with real coefficients. The operations of addition and multiplication are commutative because A+B=B+A and A⋅B=B⋅A for all matrices A and B with real coefficients. The operations of addition and multiplication are associative because (A+B)+C=A+(B+C) and (A×B)×C=A×(B×C) for all matrices A, B, and C with real coefficients. The additive identity element is the zero matrix, and the multiplicative identity element is the identity matrix. Every matrix has an additive inverse, and every nonzero matrix has a multiplicative inverse.
Commutative ring
A ring is a commutative ring if the multiplication operation is commutative for all elements in the ring. That is, if all a and b in R satisfy the property.
M4. Commutative: An operation × is commutative if a × b = b × a
Let’s take some examples to understand:
The set of polynomials with real coefficients under addition and multiplication: The set of polynomials with real coefficients is closed under addition and multiplication because the addition and multiplication of any two polynomials with real coefficients are also a polynomial with real coefficients. The operations of addition and multiplication are commutative because p(x)+q(x)=q(x)+p(x) and p(x)×q(x)=q(x)×p(x) for all polynomials p(x) and q(x) with real coefficients. The additive identity element is the polynomial 0(x), and the multiplicative identity element is the polynomial 1(x). Every polynomial has an additive inverse, and every nonzero polynomial has a multiplicative inverse.
For instance, the polynomials x²+2x+1 and x+1 product is the polynomial x³+3x²+3x+1. The product of x+1 and x²+2x+1 is also the polynomial x³+3x²+3x+1. This shows that the multiplication operation in the set of polynomials with real coefficients is commutative.
In summary, ring theory is a branch of abstract algebra that studies algebraic structures called rings. These rings are sets with two binary operations, usually addition and multiplication, that follow certain properties. Ring theory has numerous applications in number theory, coding, algebraic geometry, and computer science. It provides a robust framework for understanding algebraic structures, which underlies various applications in mathematics and related disciplines.
