Ring
RING
A ring is algebraic structure (R, '+' , '.') that consists of a set equipped with two binary operations, usually called addition and multiplication. The definition of a ring includes the following properties:
- Additive Closure: The set R is closed under addition, meaning that if you add any two elements from the set R, the result is also in the set R.
- Additive Associativity: Addition is an associative, meaning that for any elements a, b, and c in the set R s.t. (a+b)+c=a+(b+c)
- Additive Inverse: For every element a in the set R, there exists an additive inverse (usually denoted as -a) in the set R such that a+(-a)=0 and (-a)+a=0.
- Additive Identity: There exists an additive identity element (usually denoted as 0) in the set R, such that for any element a in the set R, a+0=a and 0+a=a.
- Commutative: The commutative law for addition states that for any elements a and b in a set R s.t. a+b = b+a.
- Multiplicative Closure: The set is closed under multiplication, meaning that if you multiply any two elements from the set R, the result is also in the set R.
- Multiplicative Associativity: Multiplication is an associative operation, meaning that for any elements a, b, and c in the set R s.t. (a⋅b)⋅c=a⋅(b⋅c).
- Distributive law: For any numbers a, b, and c in R s.t. a⋅(b+c)=(a⋅b)+(a⋅c).
Ring with Unity:
A ring with unity, also known as a unital ring or a ring
with identity, is a mathematical structure that satisfies all the properties of
a ring along with the existence of a multiplicative identity element (usually
denoted as 1). For any element a
in the ring, the product 1⋅a=a⋅1=a.
The presence of a multiplicative identity distinguishes a ring with unity from
a general ring.
Commutative Ring:
A commutative ring is a ring in which the multiplication
operation is commutative. In other words, for any elements a and b in the ring, a⋅b=b⋅a. The commutative property of
multiplication is an additional condition imposed on a ring.
Null Ring: The null ring, also known as the zero ring or trivial ring,
is the simplest ring. It consists of a single element, typically denoted as 0,
and has trivial operations: 0+0=0 and 0⋅0=0. The null ring is both a ring
and a commutative ring.
Ring of Integers: The ring of integers (Z) consists of all integers, both positive and negative, along with zero. The operations of addition and multiplication on integers satisfy the properties of a ring, making Z an integral domain.
Ring of Real Numbers: The ring of real numbers (R) is a set that includes all real numbers, and the operations of addition and multiplication on real numbers satisfy the properties of a ring. Real numbers also form a field, which means that every nonzero element has a multiplicative inverse.
Ring of Rational Numbers: The ring of rational numbers (Q) includes all numbers that can be expressed as fractions of integers (where the denominator is not zero). The operations of addition and multiplication on rational numbers satisfy the properties of a ring. Similar to real numbers, rational numbers form a field.
Ring of Matrices: The ring of matrices, denoted as Mn(R) or Mn(C) for matrices of size n×n with real or complex entries, respectively, is a set of matrices. The operations of matrix addition and multiplication satisfy the properties of a ring. However, not all matrices have multiplicative inverses, so the set of matrices forms a ring but not a field.
Ring with Zero Divisors
A ring with zero divisors is a mathematical structure where
there exist nonzero elements a
and b in the ring such
that their product a⋅b equals zero. In other words,
there are elements in the ring, excluding zero, whose multiplication results in
zero. Formally, if there are elements a
and b in the ring, a≠0 and b≠0, such that a⋅b=0,
then the ring is said to have zero divisors.
Ring without Zero Divisors
A ring without zero divisors, also known as an integral
domain, is a special type of ring in which the product of any two nonzero
elements is always nonzero. In an integral domain, if a and b
are elements in the ring, and a⋅b=0, then it
must be the case that either a
or b (or both) is equal
to zero. Formally, if a⋅b=0 implies
that either a=0 or b=0 (or both), then the ring
is an integral domain.
Integral Domain
An integral domain is a mathematical structure that
satisfies the properties of a commutative ring with unity and has no zero
divisors. Formally, a ring R
is an integral domain if, for any nonzero elements a and b
in R, the product a⋅b
is nonzero. In other words, if a⋅b=0, then a=0 or b=0, ensuring that there are
no nontrivial zero divisors.
Field
A field is a more advanced algebraic structure than an
integral domain. It is a set equipped with two binary operations, usually
called addition and multiplication, where every nonzero element has a
multiplicative inverse. Formally, a field F
is a set with two operations, addition and multiplication, such that:
Additive and Multiplicative Closure: For any elements a and b in F,
a+b and a⋅b is in F.
Associativity: Addition and multiplication are associative
operations.
Additive and Multiplicative Identity: There exist additive
and multiplicative identity elements (usually denoted as 0 and 1) such that a+0=a and a⋅1=a for any a in F.
Additive Inverses: For every element a in F,
there exists an additive inverse -a
such that a+(-a) =0.
Commutative: For any elements a and b in the ring, a⋅b=b⋅a.
Distributive: For any numbers a, b, and c in R s.t. a⋅(b+c)=(a⋅b)+(a⋅c).
Nonzero Multiplicative Inverses: For every nonzero element a in F, there exists a multiplicative inverse
or
A commutative ring R with unit element, having at least two elements is called a field, if every non zero elements of R possesses their multiplicative inverse
Skew Field
A skew field, also known as a division ring, is a more
general algebraic structure than a field. Unlike a field, a skew field allows
for noncommutative multiplication. Formally, a skew field D is a set equipped with two
operations, addition and multiplication, satisfying the properties of a ring
with unity (a non-zero element 1 such that 1⋅a=a⋅1=a
for all a in D) and such that every nonzero
element has both a left and a right multiplicative inverse. In other words, for
every nonzero element a in
D, there exist elements b and c in D
such that a⋅b=c⋅a=1,
allowing for noncommutative inverses.
Subring
A subring is a subset of a ring that is itself a
ring with respect to the same operations of addition and multiplication defined
on the larger ring.
Proper Subring
A subring S of a ring R is considered a proper subring if S is a subring of
R and S is not equal to R.
Improper Subring
A subring S of a ring R is considered an improper
subring if S is a subring of R and S is equal to R.
Subfield
A subfield is a subset of a field that is itself a
field with respect to the same operations of addition and multiplication
defined in the larger field.
Characteristic of a Ring
The characteristic of a ring with unity is defined as the smallest positive integer n such that n⋅1=0, where 1 is the multiplicative identity of the ring or it is the smallest positive integer n such that adding the multiplicative identity 1 to itself n times results in the additive identity 0.
Characteristic of a Field
The characteristic of a field is the smallest positive integer n such that n⋅1=0, where 1 is the
multiplicative identity of the field. If no such n exists, the characteristic
is defined to be zero.
Fields can have characteristic zero or characteristic p for some prime
number p.
Ordered Integral Domain
An ordered integral domain is an integral domain (a non-empty set equipped
with addition and multiplication operations satisfying certain properties) that
is also equipped with a total order relation (≤≤) that is compatible with
addition and multiplication.
The total order relation must satisfy:
If a≤b, then a+c≤b+c for all elements a,b,c in the domain.
If 0≤a and 0≤b, then 0≤a⋅b.
An example of an ordered integral domain is the set of real numbers.
Ordered Relation in an Integral Domain
In an integral domain, an ordered relation typically refers to a total
order relation (≤≤) that is defined on the elements of the domain.
The ordered relation is often required to satisfy certain properties,
especially compatibility with addition and multiplication, as mentioned in the
definition of an ordered integral domain.
Polynomial Rings
A polynomial ring is a mathematical structure formed by polynomials with
coefficients from a given ring. The polynomial ring is denoted as R[x], where R
is the ring of coefficients, and x is an indeterminate (variable).
The elements of the polynomial ring are polynomials, which are formal
expressions of the form anxn+an−1xn−1+…+a1x+a0,
where ai are coefficients from the ring R.
Operations in the polynomial ring involve addition and multiplication of
polynomials.
Degree of a Polynomial
The degree of a polynomial is the highest power of the indeterminate x with
a non-zero coefficient in the polynomial.
For a polynomial anxn+an−1xn−1+…+a1x+a0,
the degree is n.
The degree of the zero polynomial is often undefined or sometimes considered as negative infinity.
Left Ideal
Let R be a ring and I be a subset of R. I is called a left ideal of R if:
Additive Closure: For any elements x and y in I, the sum x+y is also in I.
Left Absorption Property: For any element r in R and any element x in I, the product rx is in I.
Right Ideal
Let R be a ring and J be a subset of R. J is called a right ideal of R if:
Additive Closure: For any elements x and y in J, the sum x+y is also in J.
Right Absorption Property: For any element r in R and any element x in J, the product xr is in J.
Ideal
Let R be a ring. An ideal I of R is a non empty subset of R with the following properties:
Additive Closure: For any elements x and y in I, the sum x + y is also in I.
Absorption Property: For any element x in I and any element r in R, both rx and xr are in I.
or
If a subset K of a ring R is both a left ideal and a right ideal, then it is called a two-sided ideal or simply an ideal of R. In other words, K is a two-sided ideal if it satisfies both left and right absorption properties.
Homomorphism
A homomorphism between two algebraic structures is a map that preserves the
operations of the structures. Let R and S be two rings. A function ϕ:R→S is a
ring homomorphism if, for all a,b∈R:
ϕ(a+b)=ϕ(a)+ϕ(b)
ϕ(a⋅b)=ϕ(a)⋅ϕ(b)
Kernel
For a ring homomorphism ϕ:R→S, the kernel of ϕ, denoted by ker(ϕ), is the
set of elements in R that map to the additive identity in S. In other words:
ker(ϕ)={r∈R , ϕ(r)=0S}
Isomorphism
A ring homomorphism ϕ:R→S is called an isomorphism if it is a bijective
(one-to-one and onto) homomorphism.
Quotient Ring
Let R be a ring and I be an ideal of R. The quotient ring, denoted by R/I, is the set of cosets {r+I∣r∈R} with addition and multiplication defined as:
(r1+I) + (r2+I) = (r1+r2) + I and (r1+I) ⋅ (r2+I) = (r1⋅r2) + I
Principle Ideal
An ideal I in a ring R is called a principal ideal if I is generated by a
single element, i.e., I=(a) for some a in R.
Principle Ideal Ring
A principal ideal ring is a type of ring in which every ideal is a principal ideal.
Ring Divisibility in an Integral Domain
In an integral domain (a commutative ring with no zero divisors), we say
that an element a divides an element b (denoted as a∣b) if there exists an
element c in the ring such that b=ac.
Prime Ideal
An ideal P in a commutative ring R is called a prime ideal if, for any
elements a and b in R, if ab is in P, then either a or b (or both) are in P.
Maximal Ideal
An ideal M in a ring R is called a maximal ideal if there is no proper
ideal I (other than R itself) such that M⊆I⊆R.