services@thehomeworkhelp.co.uk
Homework Help
Homework Help
View Details
Assignment Help
Assignment Help
View Details
Online Tutoring
Online Tutoring
View Details
Home » Math Homework Help » Algebra Homework Help » Rings of a Set
Rings of a Set
A non-empty set R together with two binary operations ‘*’ and Δ, defined in it, is said to form a ring if the following axioms are satisfied.

(a) Axioms due to the composition “*”:

R1: R is closed w.r.t. “*”, i.e. ∀ a, b R a * b R.

R2: * is associative in R i.e. a * (b * c) = (a * b) * c, ∀ a, b, c R.

R3: Identity element w.r.t. *. There exists e R such that

e * a = a * e = a, for every a R, then

R4: Existence of inverse of every element: For every a R, there exists b R such that a * b = b * a = e, then b is called the inverse of a and vice versa.

R5: * is commutative in R, i.e. a * b = b * a. ∀ a, b R.

(b) Axioms due to both the compositions * a and Δ.

R6: R is closed w.r.t. Δ, i.e. ∀ a, b R a Δ b R.

R7: Δ is associative in R, i.e. a Δ (b Δ c) = (a Δ b) Δ c, ∀ a, b, c R.

R8: Distributive Laws:

Left distributive law: a Δ (b Δ c) = (a Δ b)* (a Δ c), ∀ a, b, c R.

Right distributive law: (b * c) Δ a = (b Δ a)*(c Δ a), ∀ a, b, c R.

In other words, an algebraic structure < R, *, Δ > is said to form an associative ring if

(a) < R, * > is an algebraic group

(b) < R, Δ > is a semi-group

(c) Both right and left distributive laws hold in R.

Some special types of rings

(I) Ring with unity: A ring < R, * Δ > is said to be a ring with unity, if there exists an element g R such that

g Δ a = a = a Δ g                     ∀ a R   

then the element g is called the unity of the ring.

(II) Commutative ring: A ring < R, *, Δ > is said to be a commutative ring if Δ is commutative in R

i.e. a Δ b = b Δ a                       ∀ a, b R   

(III) Ring without zero divisors: A ring < R, *, Δ > is said to be ring without zero divisors if the product of two non-zero elements or R is zero, i.e. if ∀ a, b R,

a Δ b = 0 either a = 0 or b = 0.

(IV) Ring with zero divisors: A ring < R, *, Δ > is said to be ring with zero divisors if for some pair of elements ∀ a, b R, a ≠ 0, b ≠ 0, but, we have

a Δ b = 0.

The ring of integers with addition and multiplication module 5 is a ring with zero divisors as 3 × 57 = 0 so 3 and 7 are zero divisors.

(V) Integral domain: A ring < R, *, Δ > which is such that

(i) it is commutative, (ii) it has a unity, (iii) it is without zero divisors, is called an integral domain.

(VI) Division ring or skew field: A triplet < R, *, Δ > is said to be skew-field or division field if

(i) < R, *, Δ > is a ring with unity

(ii) For all a R ~ (0), there exists an element bR ~ (0) such that

a Δ b = b Δ a = g

In other words a R ~ (0) is invertible.

(VII) Field: A commutative division ring is called a field.

(VIII) Idempotent ring: A ring R is said to be idempotent if

x2 = x, ∀ x R

(IX) Boolean ring: A ring R is said to be Boolean ring if for every x R, x2 = x, i.e. if the ring is idempotent.

Services: - Rings of a Set Homework | Rings of a Set Homework Help | Rings of a Set Homework Help Services | Live Rings of a Set Homework Help | Rings of a Set Homework Tutors | Online Rings of a Set Homework Help | Rings of a Set Tutors | Online Rings of a Set Tutors | Rings of a Set Homework Services | Rings of a Set


Submit Your Query ???
Assignment Help
Calculus Homework Help Algebra Homework Help Abstract Algebra Help Discrete Math Help
Topics
Real Number Absolute Value Addition Of Matrices Square Matrix Adjoint Algebraic Structures Alternating Series Linear Equations Determinants Archimedean Real Numbers Binary Operation Binary Relation In A Set Bounded, Unbounded Sets Cauchy Root Test Caylay Hamiltion Theorem Circular Permutation Common Roots Complex Numbers Complex Number Conjugate Conjugate Of A Matrix Constant Sequences Convergence Of A Sequence Cosets Cubic, Biquadratic Equations De Moivre Theorem Real Number Denseness Order 3 Determinants Differences Of Matrices Direct Sum Of Vector Subspaces Eigen Vector Elementary Matrices Matrix Elem. Transformations Equal Matrices Equal Roots Two Permutations Equity Equivalent Matrices Trigonometry Function Expansion Field Function Algebra Fundamental Theorem Gaussian Integer Geometric Series Group Ideals Quantity Increasing Roots Infinite Series Convergent Integers Inverse Of Square Matrix Inverses Of Elementary Matrices Iota, Imaginary Numbers Left-Right Identity Sequence Limit Points Linear Combination Vectors Span Linear Dependence, Independence Linear Homogeneous Equations Two Subspaces Linear Sum Matric Polynomial Matrix Linear Equation Matrix Inverse Matrix Multiplication Matrix Scalar Multiplication Method Of Difference Minors And Co-factors Multiplication Modulo P Normal Sub-Group Normalizer Or Centalizer Orbit Of Permutation Peano Axioms Permutation Function Pigeon Hole Principle Matrices Integral Powers Mathematical Induction Principal Two Determinants Product Two Permutations Product Properties Of Modulus Rank Of A Matrix Rational Numbers Rational, Integral Polynomial Reciprocal Roots Relation Of Sets Rings Of A Set Row By Column Matrix Sequence Series Series Of Positive Terms Series Partial Sum Sequence Subrings Sum Of A Series Cosine Series Sum Sum Of Sine Series Symmetric/Skew Symm. Matrices Roots Symmetric Functions Symmetric Set Degree N Synthetic Division Transformation In General Transformations Of Equations Transpose Of A Matrix Matrix Transposed Conjugate Transposition Complex Numbers Representation Vector Space Vector Sub-spaces Whole Numbers