Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Idempotent Elements and Central Orthogonal

In course

MATH-215: Algebra III - rings and fields

This is an introductory course in ring and field theory. ...

Description

This lecture covers the concepts of idempotent elements, central orthogonal elements, and their properties in non-central rings. It also discusses the conditions for a ring to be commutative and the existence of prime ideals. The instructor explains the importance of idempotent elements in ring theory and provides examples to illustrate these concepts. Additionally, the lecture delves into the notion of maximal ideals and their significance in ring theory, emphasizing the relationship between integrality and maximality in a ring.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructor

Zsolt Patakfalvi

See CV here .

Official source

Ontological neighbourhood

Mathematics

Algebra: Abstract algebra

Related concepts (32)

Ideal (ring theory)

In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal.

Prime ideal

In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal. Primitive ideals are prime, and prime ideals are both primary and semiprime. An ideal P of a commutative ring R is prime if it has the following two properties: If a and b are two elements of R such that their product ab is an element of P, then a is in P or b is in P, P is not the whole ring R.

Ideal (order theory)

In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory. A subset I of a partially ordered set is an ideal, if the following conditions hold: I is non-empty, for every x in I and y in P, y ≤ x implies that y is in I (I is a lower set), for every x, y in I, there is some element z in I, such that x ≤ z and y ≤ z (I is a directed set).

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 domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed integral ideals for clarity. Let be an integral domain, and let be its field of fractions.

Idempotent (ring theory)

In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring is an element a such that a2 = a. That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that a = a2 = a3 = a4 = ... = an for any positive integer n. For example, an idempotent element of a matrix ring is precisely an idempotent matrix. For general rings, elements idempotent under multiplication are involved in decompositions of modules, and connected to homological properties of the ring.

Related lectures (33)

Integrity in Algebra

Explores the properties of commutative rings and integral domains in algebra.

Dimension Theory of Rings Log in to Mediaspace to watch this video

Explores the dimension theory of rings, focusing on chains of ideals and prime ideals.

Division Rings and Ideals Log in to Mediaspace to watch this video

Explores division rings, integral domains, fields, and ideals in rings, with examples and key theorems.

Ring Constructions: Structure Theorems Log in to Mediaspace to watch this video

Explores operations on ideals and structure theorems in commutative rings.

Chinese Remainder Theorem: Rings and Fields Log in to Mediaspace to watch this video