Main Index
Algebraic structures
Ring theory
Commutative rings
Subject Index
comment on the page
An integral domain (or an entire ring) is a commutative ring with a multiplicative identity, say
, such that
, in which the product of any two non-zero elements is always non-zero.
If and
are elements of the integral domain R, we say that
divides
or that
is a divisor of
or that
is a multiple of
if there exists an element
such that
. Symbolically we write
, or simply
.
The divisibility relation is transitive: if divides
and
divides
, then
divides
.
An integral domain can be thus defined as a commutative ring with identity and with no non-zero divisors.
The elements which divide the identity are called the units of
. These are precisely the invertible elements in
and form a subgroup of the multiplicative semigroup of non-zero elements of
. Units divide all other elements.
Moreover, if divides
, then
divides every multiple of
. If
divides two elements of
, then
also divides their sum and difference.
If divides
and
divides
, then we say
and
are associated elements. Elements
and
are associated if and only if there exists a unit
such that
.
If is a non-unit of
, we say that
is an irreducible element if
cannot be written as a product of two non-units of
.
If is a non-zero non-unit, we say that
is a prime element in
if, whenever
divides a product
,
, then
divides
or
divides
.
Cite this web-page as:
Štefan Porubský: Integral domain.