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Final Answers
© 2000-2015   Gérard P. Michon, Ph.D.

Ring Theory

The Road goes ever on and on, down from the door where it began.
The Lord of the Rings  by J.R.R. Tolkien (1892-1973).

On this site, see also:

Related Links (Outside this Site)

The development of Ring Theory  by  J.J. O'Connor  and  E.F. Robertson.
The Mathematical Atlas:  Commutative rings and algebras.

Wikipedia :   Ring Theory  |  Zero divisor  |  Ideals  |  Adele ring  |  Artinian ring
MathWorld :   Rings  |  Principal rings  |  Euclidean rings  |  Ideals.  |  Class number


Rings and Ideals

(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.

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 fieldnoncommutative 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 ringsfields  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  nth  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  Z / 4Z  (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.

Idempotent ring elements

(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, 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   =   xp + yp

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) = xp  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.

The automorphism group of the Galois field  GF(p)  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.

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, aÌ 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 Z  is the ideal consisting of all even integers).  An ideal so  "generated"  by a single element is called a  principal ideal.

A ring, like Z, 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  nilpotent  residues.

(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, Z / 4Z  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  nilpotent  divisor of zero.)
+ 0123
´ 0123

The notation  Zp  instead of  Z / pZ  is  not  recommended, as the former is best reserved for the ring of p-adic integers.

 Augustin Cauchy 
 (1789-1857) (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:

cn   =    å   ai bn-i
i = 0

Namely:  c0 = a0 b0 ,   c1 = a0 b1 + a1 b0 ,   c2 = a0 b2 + a1 b1 + a2 b0 ,  etc.

The set  A N ,  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  direct  addition  (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(  N )  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.

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 zerothe degree of a product is always the sum of the degrees.

Formal Polynomials  vs.  Polynomial Functions :

To a polynomial  (a0 , a1 ... an )  of degree  n,  we associate a function  f :

 f (x)   =    å   ai xi
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  xq  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  convergent  power 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 = 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)   =   (Z/qZ)[x] / f (x)

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

Maximal ideals .../...

Jacobson ring   |   Hilbert's Nullstellensatz

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