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...

, more specifically in ring theory

Ring theory

In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...

, a ring extension or extension ring is a ring

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...

 R with a subring

Subring

In mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R...

 S. We write R/S and say R is a ring extension of S

Given an extension R/S of commutative rings and a 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...

 P of R, it follows that
the intersection, say p, of P with S is a prime ideal of S. In this case we say that P lies over p. The situation is more complicated when R is not commutative.

See also

• Integral extension
• Group extension
  Group extension
  In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence...
  
  
• Algebraic extension
  Algebraic extension
  In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e...
  
  
• Ore extension
  Ore extension
  In mathematics, especially in the area of algebra known as ring theory, an Ore extension, named after Oystein Ore, is a special type of a ring extension whose properties are relatively well understood...