| Integral Domain |
Article Index for Integral |
Shopping Integral |
Website Links For Integral |
Information AboutIntegral Domain |
| CATEGORIES ABOUT INTEGRAL DOMAIN | |
| commutative algebra | |
| ring theory | |
|
Alternatively and equivalently, integral domains may be defined as commutative rings in which the zero Ideal {0} is Prime , or as the Subring s of Fields . Additionally, a commutative ring ''R'' is an integral domain if and only if for every element ''r'' of the ring, the ''R''-module map induced by multiplication by ''r'' is Injective (such ''r'' are called regular). Viewing the underlying commutative ring as a Preadditive Category , the above criterion on zero divisors is equivalent to the condition that every nonzero Morphism is a Monomorphism (hence also an Epimorphism , by making use of the bilinear structure on the set of morphisms). The condition 0 ≠ 1 only serves to exclude the Trivial Ring {0} with a single element. EXAMPLES
:
DIVISIBILITY, PRIME AND IRREDUCIBLE ELEMENTS If ''a'' and ''b'' are elements of the integral domain ''R'', we say that ''a divides b'' or ''a is a Divisor of b'' or ''b is a multiple of a'' if and only if there exists an element ''x'' in ''R'' such that ''ax'' = ''b''. If ''a'' divides ''b'' and ''b'' divides ''c'', then ''a'' divides ''c''. If ''a'' divides ''b'', then ''a'' divides every multiple of ''b''. If ''a'' divides two elements, then ''a'' also divides their sum and difference. The elements which divide 1 are called the ''units'' of ''R''; these are precisely the invertible elements in ''R''. Units divide all other elements. If ''a'' divides ''b'' and ''b'' divides ''a'', then we say ''a'' and ''b'' are ''associated elements''. ''a'' and ''b'' are associated if and only if there exists a unit ''u'' such that ''au'' = ''b''. If ''q'' is a non-unit, we say that ''q'' is an ''irreducible element'' if ''q'' cannot be written as a product of two non-units. If ''p'' is a non-zero non-unit, we say that ''p'' is a ''prime element'' if, whenever ''p'' divides a product ''ab'', then ''p'' divides ''a'' or ''b''. This generalizes the ordinary definition of Prime Number in the ring Z, except that it allows for negative prime elements. If ''p'' is a prime element, then the principal ideal (''p'') generated by ''p'' is a Prime Ideal . Every prime element is irreducible (here, for the first time, we need ''R'' to be an integral domain), but the converse is not true in all integral domains (it is true in Unique Factorization Domain s, however). FIELD OF FRACTIONS If ''R'' is a given integral domain, the smallest field containing ''R'' as a subring is uniquely determined up to isomorphism and is called the '' Field Of Fractions '' or ''quotient field'' of ''R''. It can be thought of as consisting of all fractions ''a''/''b'' with ''a'' and ''b'' in ''R'' and ''b'' ≠ 0, modulo an appropriate equivalence relation. The field of fractions of the integers is the field of Rational Number s. The field of fractions of a field is Isomorphic to the field itself. ALGEBRAIC GEOMETRY In Algebraic Geometry , integral domains correspond to Irreducible Varieties . They have a unique Generic Point , given by the zero ideal. Integral domains are also characterized by the condition that they are Reduced and irreducible. The former condition ensures that the nilradical of the ring is zero, so that the intersection of all the ring's minimal primes is zero. The latter condition is that the ring have only one minimal prime. It follows that the unique minimal ideal of a reduced and irreducible ring is the zero ideal, hence such rings are integral domains. The converse is clear: No integral domain can have nilpotent elements, and the zero ideal is the unique minimal ideal. CHARACTERISTIC AND HOMOMORPHISMS The Characteristic of every integral domain is either zero or a Prime Number . If ''R'' is an integral domain with prime characteristic ''p'', then ''f''(''x'') = ''x'' ''p'' defines an . SEE ALSO
|
|
|