Difference between revisions of "Integral domain"
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| − | A [[commutative ring]] with identity and without divisors of zero (cf. [[ | + | 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]]). | 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]]). | ||
Revision as of 22:15, 28 November 2014
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) |
Integral domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_domain&oldid=35064