(2006-02-15) Rings
Addition, subtraction and multiplication are defined,
division needn't be.

(A, +, . ) is a ring when
"addition" (+) and "multiplication" (.)
are well-defined internal operations
over the set A with the following properties:

(A,+) is a 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 (.).

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).
In French, a field (corps) is a division ring,
commutative or not.

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.
In a semiring, zero is postulated
to be multiplicatively absorbent (0.x = x.0 = 0).

(2006-02-15) Divisors of Zero (or zero divisor )
In some rings, the product of two nonzero elements can be zero.

In a ring, by definition, a nonzero
element d is said to be a
divisor of a given element a when there is a
nonzero element x such that:

d x = a

In particular (with a = 0 ) a divisor of zero
is a nonzero element whose product by some nonzero element is equal to zero.
There are no such things in integral domains
(including division rings, fields and skew fields).
The zero element itself is not considered a divisor of zero.

An idempotent element is a solution of the equation:

x^{ 2} = x

Every ring has at least one idempotent element (namely 0) and every
unital ring has another trivial one (namely 1).

If a unital ring has other idempotent elements (said to be nontrivial)
then it has at least one divisor of zero because,
in a unital ring, the previous equation reduces to the following zero product
of two factors (neither of which is zero when x is neither 0 nor 1).

x (1-x) = 0

If some n^{th} power of an element is zero,
that element is said to be nilpotent.
Clearly, a nonzero nilpotent element is a divisor of zero.

x^{ n} = 0

The simplest example
of a nilpotent element is "2" in the ring
/
4
(the ring formed by the 4 residues of integers
modulo 4)
where:

2^{ 2} = 0

One example of a ring with divisors of zero
but no nilpotent elements is the ring of
polyadic integers
for any radix g that's not a power of a prime.

The terms idempotent and nilpotent
were coined in 1870 by the American mathematician
Benjamin Peirce
(1809-1880). Peirce (whose name rhymes with "terse" or "purse")
taught at Harvard for nearly 50 years and is also
remembered for proving (in 1832)
that an odd perfect number
(if such a thing exists) cannot have fewer than 4 distinct prime factors.

If all the elements in this sequence are nonzero, 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 characteristic 1 is the trivial
field (where 1 = 0).

The characteristic of a nontrivial unital ring without divisors of zero
is either 0 or 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 (1880):

If the characteristic p
of a commutative ring is a prime number, we have:

( x y )^{ p} = x^{ p} y^{ p} (x + y)^{ p} = x^{p} + y^{p}

The former relation is due to 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) = x^{p} thus
respects both addition and multiplication.
It is a ring homomorphism,
which is called the Frobenius map in honor of
F. Georg Frobenius (1849-1917)
who discovered the relevance of such things to algebraic number theory, in 1880.

(2006-02-15) Ideal I in a Ring A
An ideal is a multiplicatively absorbent subring.

A subring is a ring contained in another
(using the same operations).

An_{ }ideal is a subring that 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 so "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 a PID.

Ideals were introduced in 1871 by Richard
Dedekind (1831-1916) as he investigated 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 factor
(e.g., 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 one of their powers in I.
The radical of an ideal is an ideal.
If an ideal is the radical of another it's called a radical ideal.
Every prime ideal
is a radical ideal.
Modulo a radical ideal,
there are no nilpotentresidues.

(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 just 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's a ring, 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 nilpotentdivisor 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
(a_{0} , a_{1} , a_{2} , ...) and
(b_{0} , b_{1} , b_{2} , ...)
of elements from a ring A is the sequence
(c_{0} , c_{1} , c_{2} , ...) where:

The set
A^{ },
of the sequences whose terms are elements of the ring A
has the structure of a ring (dubbed
formal power series over A)
if endowed with directaddition (the n-th term of a sum being the sum of the n-th terms of the
two addends) and the
Cauchy multiplication defined above.

The set 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 formal polynomials over a ring A It's endowed with
component-wise addition and Cauchy multiplication.

A finite sequence of elements of a ring A
(or, equivalently, a sequence with finitely many nonzero elements)
is called a polynomial over A.
The set of all such polynomials is a ring
(often denoted denoted A[x]
where x is a "dummy variable")
which is a subring of the aforementioned ring of
"formal power series",
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.

Formal Polynomials vs. Polynomial Functions :

To a polynomial
(a_{0} , a_{1} ... a_{n} )
of degree n, we associate a
functionf :

n

f (x) =

å

a_{i} x^{i}

i = 0

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 x^{q}
correspond to the same function.

In other words, the map from polynomials to
polynomial functions
need not be injective.
However, that map is indeed injective in the special case of polynomials over
ordinary signed integers or any superset thereof,
including rational, real, surreal or complex numbers
(and p-adic numbers too, for good measure).

Whenever the distinction between a polynomial and its
associated function must be stressed,
the former may be called a formal polynomial.

Similarly, infinite sequences of coefficients
are called formal power series
Unlike polynomials,
those may or may not be associated with a convergentpower series which would define a proper function...

Over a noncommutative ring, the concept of polynomials does not break down, but
the above association of a polynomial with a function is dubious.

(2006-04-05) GR(q,r) :
Galois ring of characteristic q = p^{m} 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 monicpolynomial
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)

(2014-12-06) Hilbert rings = Jacobson rings
In those, every prime ideal is an intersection of primitive ideals.