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Lecture

Congruence Relations in Rings

In course

MATH-310: Algebra

This is an introduction to modern algebra: groups, rings and fields. ...

Description

This lecture covers congruence relations in a ring, the relationship between ideals and congruence relations in a commutative ring, the ring structure on the set of congruence classes, principal ideals, principal ideal domains (PID), ring homomorphisms, the kernel of a ring homomorphism, the image as a subring, homomorphisms between Z/nZ and Z/mZ, the characteristic of a ring, and the characteristic of a direct product of rings.

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Instructor

Anna Lachowska

Official source

Ontological neighbourhood

Mathematics

Algebra: Abstract algebra

Related concepts (32)

Ring (mathematics)

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

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.

Principal ideal domain

In mathematics, 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. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors (e.g., Bourbaki) refer to PIDs as principal rings. The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot.

Principal ideal ring

In mathematics, a principal right (left) ideal ring is a ring R in which every right (left) ideal is of the form xR (Rx) for some element x of R. (The right and left ideals of this form, generated by one element, are called principal ideals.) When this is satisfied for both left and right ideals, such as the case when R is a commutative ring, R can be called a principal ideal ring, or simply principal ring. If only the finitely generated right ideals of R are principal, then R is called a right Bézout ring.

Principal ideal

In mathematics, specifically ring theory, a principal ideal is an ideal in a ring that is generated by a single element of through multiplication by every element of The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset generated by a single element which is to say the set of all elements less than or equal to in The remainder of this article addresses the ring-theoretic concept.

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