In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, especially in the area of algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

 known as commutative algebra

Commutative algebra

Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...

, certain prime ideal

Prime ideal

In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...

s called minimal prime ideals play an important role in understanding rings

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

 and modules

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

. The notion of height and Krull's Hauptidealsatz use minimal primes.

Definition

A prime ideal P is said to be a minimal prime ideal over an ideal I if there are no prime ideals strictly contained in P that contain I. A prime ideal is said to be a minimal prime ideal if it is a minimal prime ideal over the zero ideal.

Examples

• In a commutative artinian ring
  Artinian ring
  In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
  
  , every maximal ideal
  Maximal ideal
  In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...
  
   is a minimal prime ideal.
• In an integral domain, the only minimal prime ideal is the zero ideal.
• In the ring Z of integer
  Integer
  The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
  
  s, the minimal prime ideals over a nonzero principal ideal
  Principal ideal
  In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R.More specifically:...
  
   (n) are the principal ideals (p), where p is a prime divisor of n. The only minimal prime ideal over the zero ideal is the zero ideal itself. Similar statements hold for any principal ideal domain
  Principal ideal domain
  In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
  
  .

Properties

All rings are assumed to be unital

Unit ring

In mathematics, a unit ring or ring with a unit is a unital ring, i.e. a ring R with a multiplicative unit element, denoted by 1R or simply 1 if there is no risk of confusion.- Alternative definitions of a ring :...

.

• Every proper ideal I in a commutative ring has at least one minimal prime ideal above it. The proof of this fact uses Zorn's lemma
  Zorn's lemma
  Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states:Suppose a partially ordered set P has the property that every chain has an upper bound in P...
  
   . Any maximal ideal
  Maximal ideal
  In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...
  
   containing I is prime, and such ideals exist, so the set of prime ideals containing I is non-empty. The intersection of a decreasing chain of prime ideals is prime. Therefore, the set of prime ideals containing I has a minimal element, which is a minimal prime over I.
• Emmy Noether
  Emmy Noether
  Amalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...
  
   showed that in a commutative Noetherian ring
  Noetherian ring
  In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
  
  , there are only finitely many minimal prime ideals over any given ideal. , .
• The radical
  Radical of an ideal
  In commutative ring theory, a branch of mathematics, the radical of an ideal I is an ideal such that an element x is in the radical if some power of x is in I. A radical ideal is an ideal that is its own radical...
  
    of any proper ideal I coincides with the intersection of the minimal prime ideals over I. .
• The set of zero divisor
  Zero divisor
  In abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. Similarly, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0. An element that is both a left and a right zero divisor is simply...
  
  s of a given commutative ring contains the union of the minimal prime ideals .
• Krull's Hauptidealsatz describes important properties of minimal prime ideals.