(2006-02-15) Rings
Addition, subtraction and multiplication are defined,
division need not be.
(A, +, . ) is a ring when
"addition" (+) and "multiplication" (.)
are two well-defined internal operations
over the set A with the following properties:
- (A,+) is an Abelian (i.e., commutative)
group. The neutral element is 0.
- Multiplication is an assocative (not necessarily commutative) operation
distributive over addition:
x.(y+z) = x.y + x.z and (x+y).z = x.z + y.z
Multiplicative notations
allow the omission of the "dot" symbol (.) in equations.
The concept of a ring was introduced by Richard Dedekind.
The name (Zahlring in German) was coined by
David Hilbert in 1897.
Axiomatic definitions were given by
Adolf Fraenkel in 1914.
Some additional properties of a ring are indicated by specific terms:
- Commutative Ring : Multiplication is commutative: x.y = y.x
- Unital Ring :
There's a multiplicative neutral element:
1.x = x.1 = x
- Integral Domain :
The product of two nonzero elements is nonzero.
- Division Ring :
Any nonzero element has a multiplicative inverse.
A field is normally defined as a commutative
division ring (a division ring where multiplication is commutative)
unless otherwise specified.
We consider as synonymous the terms noncommutative field,
noncommutative division ring and skew field
(some authors allow commutativity in a skew field).
A divisor of zero
is a nonzero element whose product by some other nonzero element is equal to zero.
There are no such things in an integral domain.
For the record, a semiring has fewer properties than a ring,
as it's built on an additive monoid
instead of an additive group.
This means that a semiring does contain a zero element
(neutral for addition) but subtraction is not always defined.
Zero is postulated to be multiplicatively absorbent
(0.x = x.0 = 0).
(2006-06-13) Characteristic of a Ring A
The smallest positive p, if any, for which all sums of p like terms vanish.
In a unital ring A,
we may call "1" the neutral element for
multiplication and name the elements of the following sequence after integers:
1, 2 = 1+1, 3 = 1+1+1, 4 = 1+1+1+1 ... (n+1) = n+1 ...
If all the elements in this sequence are nonzero, then the ring is said to have
zero characteristic.
Otherwise, the vanishing integers are multiples of the least of them,
which is called the characteristic of the ring, denoted char(A).
The only ring of characteric 1 is the trivial
field (having only one element 1 = 0).
If a unital ring has no divisors of zero, then its characteristic
is necessarily a prime number. (HINT:
any "integer" (1+1+...) corresponding to a prime divisor of a composite characteristic is a
divisor of zero.)
In particular, the characteristic of any nontrivial
field (or skew-field)
is either 0 or a prime number.
The characteristic
of a non-unital ring is defined as the least positive integer
p such that a sum of p identical terms always vanishes
(if there's no such p, then the ring is said to have zero characteristic).
Frobenius Map :
If the characteristic p
of a commutative ring is a prime number, we have:
( x y ) p = x p y p
(x + y) p = xp + yp
The former relation holds because of commutativity.
The latter relation comes from Newton's binomial formula, with the added remark
that the binomial coefficient C(p,k)
is divisible by p, if p is prime, unless k is 0 or p.
The map defined by F(x) = xp thus
respects both addition and multiplication.
It is a ring homomorphism,
which is called the Frobenius map in honor of
Georg Ferdinand Frobenius (1849-1917)
who discovered the relevance of such things to algebraic number theory, in 1880.
The automorphism group of the
Galois field
GF(pn) is a cyclic group of
order n, generated by the above Frobenius map.
(2006-02-15) Ideal I in a Ring A
An ideal is a multiplicatively absorbent subring.
A subring is a ring contained in another, endowed with
the same operations.
An ideal is a subring which contains
a product whenever it contains a factor:
For a right ideal I,
the product xa is in I whenever x is:
"aÎA,
Ia Ì I
For a left ideal I,
the product ax is in I whenever x is:
"aÎA,
aI Ì I
Unless otherwise specified,
an ideal is both a right ideal and a left ideal.
The sum, the product or the intersection of two ideals is itself an ideal
(the product of two ideals is contained in their intersection).
The sum (or the product) of two sets is defined to be the set whose elements are
sums (or products) of elements from those two sets.
One example of an ideal is the set a A of all
the multiples of an element a in the ring A
(e.g., 2 
is the ideal consisting of all even integers).
An ideal which is thus "generated" by a single element is called
a principal ideal.
A ring, like ,
whose ideals are all principal is a
principal ring.
Such a ring is called a principal integral domain
(abbreviated PID) if it has no divisors of zero
(i.e., the product of two nonzero elements is never zero).
Following Bourbaki, some authors define
a principal ring to be what we call a PID.
Ideals were introduced in 1871 by
Richard
Dedekind (1831-1916) as he considered, in particular, what are now known as
prime ideals: An ideal is defined to be prime if it doesn't contain a
product unless it contains at least one of the factors
(among integers, the multiples of a prime number form a prime ideal).
The radical Rad(I) of an ideal I
is the set of all ring elements
which have at least one of their powers in I.
The radical of an ideal is an ideal.
An ideal which is the radical of another is called a radical ideal.
In particular, every prime ideal
is a radical ideal.
There's no nilpotent residue modulo a radical ideal.
(2006-02-15) Residue Ring
(modulo a given ideal I of a ring A)
The ring A/I, which consists of all residue classes modulo I.
Modulo an ideal I of a ring A,
the residue-class (or simply the residue) [x] of an element x of A
is the set of all elements y of A
for which x-y is in I.
The set of all residues modulo I is denoted A/I.
It is a ring, which is variously called quotient ring, factor ring,
residue-class ring or simply residue ring.
|
For example, /
4 is the ring formed by
the four residue classes modulo 4, whose addition
and multiplication tables are shown at right.
(Note that "2" is a divisor of zero.)
|
| + |
0 | 1 | 2 | 3 |
| 0 | 0 | 1 | 2 | 3 |
| 1 | 1 | 2 | 3 | 0 |
| 2 | 2 | 3 | 0 | 1 |
| 3 | 3 | 0 | 1 | 2 |
|
| ´ |
0 | 1 | 2 | 3 |
| 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 2 | 3 |
| 2 | 0 | 2 | 0 | 2 |
| 3 | 0 | 3 | 2 | 1 |
|
The notation p
instead of /
p
is not recommended, as the former is best reserved for the ring of
p-adic integers.
(2006-04-27) Cauchy Product
A well-defined internal operation among sequences in a ring.
The Cauchy product of two sequences
(a0 , a1 , a2 , ...) and
(b0 , b1 , b2 , ...)
of elements from a ring A is the sequence
(c0 , c1 , c2 , ...) where:
Namely:
c0 = a0 b0 ,
c1 = a0 b1 +
a1 b0 ,
c2 = a0 b2 +
a1 b1 +
a2 b0 , etc.
The set,
denoted A
,
of the sequences whose terms are elements of the ring A
has the structure of a ring (the so-called
formal power series over A)
if endowed with direct
addition (the n-th term of a sum being the sum of the n-th terms of the
two summands) and the
Cauchy multiplication defined above.
The set which is
denoted A( )
consists of those sequences which have only finitely many nonzero terms.
It forms a subring of the above ring,
better known as the [univariate]
polynomials over A, denoted A[x]
and discussed next.
(2006-04-06) A[x] :
Ring of Polynomials over a Ring A
Component-wise addition and Cauchy multiplication.
Polynomials are just another name for finite sequences of elements
of the ring (or, alternately, sequences with only finitely many
nonzero terms). They form a subring of the above ring,
under direct addition and
Cauchy multiplication.
Each term of the sequence defining a polynomial is called a
coefficient.
The degree of a polynomial is the highest of the ranks
of its nonzero coefficients (the lowest rank being zero).
The null polynomial ("zero") has no nonzero coefficients,
and its degree
is defined to be -¥ ("minus infinity")
so that, in a ring without
divisors of zero,
the degree of a product is always the sum of the degrees.
To a polynomial of degree n
(a0 , a1 ... an )
is associated a function f
However, that function and the polynomial which defines it
are two different things entirely...
For example, over the finite field GF(q),
the distinct polynomials x and xq
correspond to the same function.
Over a noncommutative ring, the concept of polynomials does not break down, but
the above association of a polynomial with a function is questionable.
(2006-04-05) GR(q,r) :
Galois ring of characteristic q = pm and rank r
The modulo-q polynomials modulo
an irreducible polynomial modulo p.
Let q be a power of a prime p. Let f
be some monic polynomial
modulo q, of degree r, which is irreducible modulo p
(i.e., f (x) mod p never vanishes).
The Galois ring of characteristic q and rank r is
defined as:
GR(q,r) =
(/q)[x] / f (x)
