View source for Integral domain

{{TEX|done}}{{MSC|13G05}}

''integral ring''

A [[commutative ring]] with identity and without divisors of zero (cf. [[Zero divisor]]). Any field, and also any ring with identity contained in a field, is an integral domain. Conversely, an integral domain can be imbedded in a field. Such an imbedding is given by the construction of the [[field of fractions]].

If $A$ is an integral domain, then the ring of polynomials $A[X]$ and the ring of formal power series $A[[X]]$ over $A$ are also integral domains. If $A$ is a commutative ring with identity and $I$ is any ideal in $A$, then the ring $A/I$ is an integral domain if and only if $I$ is a [[Prime ideal|prime ideal]]. A ring $A$ without nilpotents is an integral domain if and only if the spectrum of $A$ is an [[irreducible topological space]] (cf. [[Spectrum of a ring|Spectrum of a ring]]).

Sometimes commutativity of $A$ is not required in the definition of an integral domain. Skew-fields and subrings of a skew-field containing the identity are examples of non-commutative integral domains. However, it is not true, in general, that an arbitrary non-commutative integral domain can be imbedded in a skew-field (see [[#References|[2]]], and [[Imbedding of rings]]).

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,   "Algebra" , Addison-Wesley  (1965)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  P.M. Cohn,   "Free rings and their relations" , Acad. Press  (1985)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  C. Faith,   "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)</TD></TR>
</table>