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Let P be the set of prime ideals of a commutative ring A. Then an affine scheme is a technical mathematical object defined as the ring spectrum sigma(A) of P, regarded as a ...

Given an affine variety V in the n-dimensional affine space K^n, where K is an algebraically closed field, the coordinate ring of V is the quotient ring ...

A homogeneous ideal I in a graded ring R= direct sum A_i is an ideal generated by a set of homogeneous elements, i.e., each one is contained in only one of the A_i. For ...

The notion of height is defined for proper ideals in a commutative Noetherian unit ring R. The height of a proper prime ideal P of R is the maximum of the lengths n of the ...

If A and B are commutative unit rings, and A is a subring of B, then A is called integrally closed in B if every element of B which is integral over A belongs to A; in other ...

A commutative Noetherian unit ring having only finitely many maximal ideals. A ring having the same properties except Noetherianity is called quasilocal. If K is a field, the ...

An algebraic variety is a generalization to n dimensions of algebraic curves. More technically, an algebraic variety is a reduced scheme of finite type over a field K. An ...

A prime ideal is an ideal I such that if ab in I, then either a in I or b in I. For example, in the integers, the ideal a=<p> (i.e., the multiples of p) is prime whenever p ...

Algebraic geometry is the study of geometries that come from algebra, in particular, from rings. In classical algebraic geometry, the algebra is the ring of polynomials, and ...

An operation on rings and modules. Given a commutative unit ring R, and a subset S of R, closed under multiplication, such that 1 in S, and 0 not in S, the localization of R ...