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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
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Elementary Matrices
Matrix Elem. Transformations
Equal Matrices
Equal Roots
Two Permutations Equity
Equivalent Matrices
Trigonometry Function Expansion
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Algebra Fundamental Theorem
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Quantity Increasing Roots
Infinite Series Convergent
Integers
Inverse Of Square Matrix
Inverses Of Elementary Matrices
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Left-Right Identity
Sequence Limit Points
Linear Combination Vectors Span
Linear Dependence, Independence
Linear Homogeneous Equations
Two Subspaces Linear Sum
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Matrix Multiplication
Matrix Scalar Multiplication
Method Of Difference
Minors And Co-factors
Multiplication Modulo P
Normal Sub-Group
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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
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Matrix Transposed Conjugate
Transposition
Complex Numbers Representation
Vector Space
Vector Sub-spaces
Whole Numbers
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