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In algebra, a domain is a nonzero ring in which ab = 0 implies a = 0 or b = 0. [1] (Sometimes such a ring is said to "have the zero-product property".) Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor). A commutative domain is called an integral domain.
The term domain is also commonly used in a different sense in mathematical analysis: a domain is a non-empty connected open set in a topological space. In particular, in real and complex analysis , a domain is a non-empty connected open subset of the real coordinate space R n {\displaystyle \mathbb {R} ^{n}} or the complex coordinate space C n ...
If A, B are integral domains over an algebraically closed field k, then A ⊗ k B is an integral domain. This is a consequence of Hilbert's nullstellensatz , [ a ] and, in algebraic geometry, it implies the statement that the coordinate ring of the product of two affine algebraic varieties over an algebraically closed field is again an integral ...
A Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal (a proof is given at the end). Also, a Dedekind domain is a UFD if and only if its ideal class group is trivial. In this case, it is in fact a principal ideal domain. In general, for an integral domain A, the following conditions are equivalent: A is a UFD.
with domain, the range of , sometimes denoted or (), [4] may refer to the codomain or target set (i.e., the set into which all of the output of is constrained to fall), or to (), the image of the domain of under (i.e., the subset of consisting of all actual outputs of ). The image of a function is always a subset of the codomain of the ...
A Euclidean domain is an integral domain which can be endowed with at least one Euclidean function. A particular Euclidean function f is not part of the definition of a Euclidean domain, as, in general, a Euclidean domain may admit many different Euclidean functions.
Similarly, when dealing with complex variables, a critical point is a point in the function's domain where its derivative is equal to zero (or the function is not holomorphic). [3] [4] Likewise, for a function of several real variables, a critical point is a value in its domain where the gradient norm is equal to zero (or undefined). [5]
In complex analysis, a complex domain (or simply domain) is any connected open subset of the complex plane C. For example, the entire complex plane is a domain, as is the open unit disk, the open upper half-plane, and so forth. Often, a complex domain serves as the domain of definition for a holomorphic function.