nonzero ideal

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• Ideal norm — In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension. It is particularly important in number theory since it measures the size of an ideal of a complicated number ring in terms of an… …   Wikipedia

• Ideal (ring theory) — In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like even number or multiple of 3 . For instance, in… …   Wikipedia

• Ideal class group — In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field (or more generally any Dedekind domain) can be described by a certain group known as an ideal class group (or class group). If… …   Wikipedia

• Norm of an ideal — The norm of an ideal is defined in algebraic number theory. Let Ksubset L be two number fields with rings of integers O Ksubset O L. Suppose that the extension L/K is a Galois extension with :G= extstyle{Gal}(L/K). The norm of an ideal I of O L… …   Wikipedia

• Prime ideal — 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. This article only covers ideals of ring theory. Prime ideals in order theory are treated in the article on… …   Wikipedia

• Maximal ideal — In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal (with respect to set inclusion) amongst all proper ideals.[1][2] In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and… …   Wikipedia

• 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.Principal ideal domains are thus mathematical objects which behave somewhat like the… …   Wikipedia

• Fractional ideal — In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral… …   Wikipedia

• Radical of an ideal — In ring theory, a branch of mathematics, the radical of an ideal is a kind of completion of the ideal. There are several special radicals associated with the entire ring such as the nilradical and the Jacobson radical , which isolate certain bad… …   Wikipedia

• 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: * a left principal ideal of R is a subset of R of the form R a := { r a : r in R }; * a… …   Wikipedia

• Dedekind domain — In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily …   Wikipedia