In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull (1899 - 1971), is defined to be the number of strict inclusions in a maximal chain of prime ideals. We take the supremum of chain lengths if no maximal chain can be found. For example, in the ring (Z/8Z)[x,y,z] we can consider the chain In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... Wolfgang Krull (1899 - 1971) was a German mathematician, after whom Krull dimension, the Krull topology, and Krulls principal ideal theorem are named. ... 1899 was a common year starting on Sunday (see link for calendar). ... 1971 is a common year starting on Friday (click for link to calendar). ... In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ... In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ...

(2) ⊂ (2,x) ⊂ (2,x,y) ⊂ (2,x,y,z)

Each of these ideals is prime, so the Krull dimension of (Z/8Z)[x, y, z] is at least the number of strict inclusions in this chain, that is, 3. In fact the dimension of this ring is exactly 3.

An alternate way of phrasing this definition is to say that the Krull dimension of R is the largest height of any prime ideal of R. In commutative algebra, the height of an ideal I in a ring R is the number of strict inclusions in the longest chain of prime ideals contained in I. In a Noetherian ring, Krulls height theorem says that the height of an ideal generated by n elements is no...

According to this convention, a integral domain of dimension zero is a field. Dedekind domains and discrete valuation rings have dimension one. In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, a Dedekind domain is a Noetherian integral domain which is integrally closed in its fraction field and which has Krull dimension 1. ... In mathematics, a discrete valuation ring (DVR) is a particular kind of commutative ring that is a local ring, which satisfies conditions that in algebraic geometry come from non-singularity of a point on an algebraic curve. ...

If a ring R has Krull dimension k, then the polynomial ring R[x] will have dimension between k + 1 and 2k + 1. If R is Noetherian, then the dimension of R[x] will be exactly k + 1. In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ...