In commutative algebra, the spectrum of a ring R is the set of all prime ideals of R, and is usually denoted by Spec R. In algebraic geometry, it is simultaneously a topological space equipped with the sheaf of rings O.
Conditions: R is a commutative ring.; The context includes both commutative algebra and algebraic geometry.
In commutative algebra, the spectrum of a ring R is the set of all prime ideals of R, and is usually denoted by Spec R. In algebraic geometry, it is simultaneously a topological space equipped with the sheaf of rings O.
Conditions: R is a commutative ring.; The context includes both commutative algebra and algebraic geometry.
An affine scheme is a locally ringed space that is isomorphic to the spectrum Spec(R) of a commutative ring R. General schemes are constructed by gluing together open sets of this simple kind, called affine schemes.
Conditions: The object is an affine scheme.; It is a locally ringed space.; It is isomorphic to the spectrum of a commutative ring.
An affine scheme is a locally ringed space that is isomorphic to the spectrum Spec(R) of a commutative ring R. General schemes are constructed by gluing together open sets of this simple kind, called affine schemes.
Conditions: The object is an affine scheme.; It is a locally ringed space.; It is isomorphic to the spectrum of a commutative ring.
An ideal P of a commutative ring R is prime if it satisfies two properties: first, if the product ab of two elements a and b in R is in P, then at least one of a or b must be in P; second, P is not the entire ring R.
Conditions: R is a commutative ring.; P is an ideal of R.; For all a, b in R, if ab is in P, then a is in P or b is in P.; P is not equal to R.
An ideal P of a commutative ring R is prime if it satisfies two properties: first, if the product ab of two elements a and b in R is in P, then at least one of a or b must be in P; second, P is not the entire ring R.
Conditions: R is a commutative ring.; P is an ideal of R.; For all a, b in R, if ab is in P, then a is in P or b is in P.; P is not equal to R.