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Rings of a Set

A non-empty set

In other words, an algebraic structure

then the element

i.e.

The ring of integers with addition and multiplication module

In other words

**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**“*”:****R**is closed w.r.t._{1}: R**“*”**, i.e.**∀ a, b R a * b R**.**R**is associative in_{2}: ***R**i.e.**a * (b * c) = (a * b) * c, ∀ a, b, c R**.**R**Identity element_{3}:**w.r.t. ***. There exists**e R**such that**e * a = a * e = a**, for every**a R**, then**For every**

RR

_{4}: Existence of inverse of every element:**a R**, there exists**b R**such that**a * b = b * a = e**, then**b**is called the inverse of**a**and vice versa.**R**is commutative in_{5}: ***R**, i.e.**a * b = b * a. ∀ a, b R**.**(b)**Axioms due to both the compositions*** a**and**Δ**.**R**is closed w.r.t._{6}: R**Δ**, i.e.**∀ a, b R a Δ b R**.**R**is associative in_{7}: Δ**R**, i.e.**a Δ (b Δ c) = (a Δ b) Δ c, ∀ a, b, c R**.**R**_{8}: 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**.**A ring**

(IV) Ring with zero divisors:(IV) Ring with zero divisors:

**< 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**x**^{2}= x, ∀ x R**(IX) Boolean ring:**A ring**R**is said to be Boolean ring if for every**x R, x**, i.e. if the ring is idempotent.^{2}= x**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 SetSubmit Your Query ???

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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
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Function
Algebra Fundamental Theorem
Gaussian Integer
Geometric Series
Group
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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

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