Namespaces
Variants
Actions

Integral domain

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

2020 Mathematics Subject Classification: Primary: 13G05 [MSN][ZBL]

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. 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).

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 [2], and Imbedding of rings).

References

[1] S. Lang, "Algebra" , Addison-Wesley (1965)
[2] P.M. Cohn, "Free rings and their relations" , Acad. Press (1985)
[3] C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973)
How to Cite This Entry:
Integral domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_domain&oldid=39096
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article