Ideals (ring theory)

In ring theory, which is a branch of abstract algebra, an **ideal** is a specific subset of a ring that has particular properties allowing it to be used in the construction of quotient rings and in the study of ring homomorphisms. ### Definition: Let \( R \) be a ring (with unity, but this requirement can be relaxed in some contexts).

In abstract algebra, particularly in the context of ring theory, a **prime ideal** is a special type of ideal that has important properties related to the structure of rings.

The Ascending Chain Condition (ACC) on principal ideals is a property related to the structure of a ring in the context of ideal theory. Specifically, a ring \( R \) satisfies the ACC on principal ideals if any ascending chain of principal ideals eventually stabilizes.

In the context of algebra, particularly in ring theory and module theory, an **augmentation ideal** is a specific ideal associated with a group ring or a similar algebraic structure. ### Definition 1. **Group Ring Context**: If \( k \) is a field and \( G \) is a group, the group ring \( k[G] \) consists of formal sums of elements of \( G \) with coefficients in \( k \).

In the context of algebraic number theory, a **fractional ideal** is a generalization of the notion of an ideal in a ring. Specifically, fractional ideals are particularly useful in the study of Dedekind domains and more generally in the structure of arithmetic in number fields. ### Definitions and Properties 1. **Integral Domain**: First, consider a domain \( R \), typically a Dedekind domain, which is an integral domain where every nonzero proper prime ideal is maximal.

In order theory, an **ideal** is a specific subset of a partially ordered set (poset) that captures a certain type of "lower" structure.

The ideal class group is an important concept in algebraic number theory, particularly in the study of ring theory and algebraic integers. It provides a way to measure the failure of unique factorization in the ring of integers of a number field.

In algebra, particularly in the context of commutative rings, the term "ideal quotient" refers to a concept that is used to define the relationship between ideals.

The Jacobian ideal is a concept in algebraic geometry and commutative algebra, associated with a polynomial ring and its derivatives. It is particularly important in the study of algebraic varieties and singularities.

Krull's theorem is a result in commutative algebra that pertains to the structure of integral domains, specifically regarding the heights of prime ideals in a Noetherian ring. The theorem states: In a Noetherian ring (or integral domain), the height of a prime ideal \( P \) is less than or equal to the number of elements in any generating set of the ideal \( P \).

In the context of ring theory, a nil ideal is an ideal \( I \) of a ring \( R \) such that every element \( x \) in \( I \) is nilpotent.

In abstract algebra, specifically in the study of rings, a **nilpotent ideal** is an ideal such that there exists some positive integer \( n \) for which the \( n \)-th power of the ideal is equal to the zero ideal.

In algebra, particularly in commutative algebra, the radical of an ideal is a fundamental concept used to study the properties of ideals and rings.

The term "real radical" can refer to a few different concepts depending on the context, but in general mathematics and algebra, a "radical" typically refers to an operation that involves roots, such as square roots, cube roots, etc. When we say "real radical," we are usually indicating that we are dealing specifically with real numbers rather than complex numbers. For example: - A real radical of the form \(\sqrt{x}\) is defined for non-negative real numbers \(x\).