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

Lecture

Rings: Operations and Ideals

In course

MATH-215: Algebra III - rings and fields

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

Description

This lecture covers the operations and properties of rings, focusing on ideals and quotients. It explains the construction of ideals, bilateral ideals, and their applications in building classes. The lecture also delves into the standard theorems related to rings and their quotients.

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)

Gödel's incompleteness theorems

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.

Non-standard model of arithmetic

In mathematical logic, a non-standard model of arithmetic is a model of (first-order) Peano arithmetic that contains non-standard numbers. The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, .... The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934).

Theorem

In mathematics, a theorem is a statement that has been proved, or can be proved. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic.

Gödel's completeness theorem

Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. The completeness theorem applies to any first-order theory: If T is such a theory, and φ is a sentence (in the same language) and every model of T is a model of φ, then there is a (first-order) proof of φ using the statements of T as axioms. One sometimes says this as "anything universally true is provable".

Tarski's undefinability theorem

Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1933, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that "arithmetical truth cannot be defined in arithmetic". The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system.

Related lectures (56)

Polynomials, Division, and Ideals

Explores polynomials, their operations, and the concept of ideals in polynomial rings.

Ideals: Polynomials and Definitions

Explores ideals in K[X], including PGCD, uniqueness, coprimality, and theorems of Bézout and Gauss.

Ring Operations: Ideals and Classes Log in to Mediaspace to watch this video

Covers the operations in rings, ideals, classes, and quotient rings.

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

Explores operations on ideals and structure theorems in commutative rings.

Regular Implicit Functions Log in to Mediaspace to watch this video