A commutative unital ring is termed an integral domain (sometimes ... A commutative unital ring R is termed an integral domain if R satisfies ... ...
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The spectrum of an integral domain is an irreducible space: it cannot ... but the ring is not in general an integral domain.
More generally ... ...
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An integral domain in which every irreducible is prime is an integral ... * Integral domain in which any two irreducible factorizations are ... ...
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Suppose A is an integral domain, and I is an ideal of A. Consider ... Given: A is an integral domain, and I is an ideal of A. N is a nonzero ... ...
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In an integral domain, any principal ideal is isomorphic, as a module ... Let A be an integral domain and I a principal ideal in A, generated ... ...
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Any regular local ring is an integral domain. In other words, if a ... must actually be a field, and hence an integral domain. ... ...
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Suppose R is a fact about::integral domain and a,b \in R are fact ... Given: An integral domain R. Two associate elements a,b \in R. ... ...
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An integral domain in which any two irreducible factorizations are ... | Weaker than::Euclidean domain || || || || integral domain in which ... ...
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Suppose R is an fact about::integral domain that is also a fact about::ring ... Given: An integral domain R satisfying the ascending chain condition ... ...
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An integral domain in which every irreducible is prime is an integral ... * Integral domain in which any two irreducible factorizations are ... ...
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* Principal ideal is isomorphic to integral domain as a module: Any principal ideal in an integral domain is isomorphic to the integral domain ... ...
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A unique factorization domain is an integral domain where every element ... A gcd domain is an integral domain for which, given any two elements ... ...
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A commutative unital ring is termed an integral domain (sometimes ... A commutative unital ring R is termed an integral domain if R satisfies ... ...
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Suppose A is an integral domain, and I is an ideal of A. Consider ... Given: A is an integral domain, and I is an ideal of A. N is a nonzero ... ...
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An integral domain is termed an adequate domain if it is also an adequate ... An integral domain R is termed an adequate domain if it satisfies ... ...
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Suppose A is a normal domain i.e. an integral domain that is integrally closed inside its fraction field. Suppose B is an integral extension ... ...
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An integral domain in which any two irreducible factorizations are ... | Weaker than::Euclidean domain || || || || integral domain in which ... ...
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An integral domain is termed a Noetherian domain if every ideal in ... * Integral domain
* Noetherian ring
==Metaproperties== ... ...
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===Integral domain===
When R has characteristic two, we have x^2 + ... field of characteristic not equal to two is integral domain ... ...
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An integral domain R is said to be a n-stage Euclidean domain if there exists a function N from the set of nonzero elements of R to the set of ... ...
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i.e. a Noetherian ring that is also an integral domain) and I a proper ideal ... ===Applying the integral domain condition and properness of the ideal=== ... ...
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A local domain is a local ring which is also an integral domain. ... * Integral domain ... ...
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This category lists properties that can be evaluated for an integral domain. Here, an integral domain is understood to be a commutative unital ... ...
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Let R be an integral domain and let K be its field of fractions. The ... * An interpolation domain is an integral domain for which interpolation ... ...
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An integral domain R is termed an interpolation domain if, for any nonnegative integer n, there exist elements (r_0, r_1, \dots, r_n) \in R^{n ... ...
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Suppose R is an integral domain with a fact about::Euclidean norm ... Given: Integral domain R, Euclidean norm N: R \setminus \{ 0 \} \to ... ...
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An integral domain R is termed a Japanese ring if the following two conditions hold:
* R is finitely generated over integers or over a field ... ...
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An integral domain is termed a gcd domain if it satisfies the following equivalent conditions:
* Given any finite collection of nonzero elements ... ...
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An fact about::irreducible element in an integral domain need not ... An integral domain in which every irreducible is prime is an integral ... ...
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