Ring homomorphism

Morphisms | Ring theory

In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is a function f : R → S such that f is: addition preserving: for all a and b in R,multiplication preserving: for all a and b in R,and unit (multiplicative identity) preserving:. Additive inverses and the additive identity are part of the structure too, but it is not necessary to require explicitly that they too are respected, because these conditions are consequences of the three conditions above. If in addition f is a bijection, then its inverse f−1 is also a ring homomorphism. In this case, f is called a ring isomorphism, and the rings R and S are called isomorphic. From the standpoint of ring theory, isomorphic rings cannot be distinguished. If R and S are rngs, then the corresponding notion is that of a rng homomorphism, defined as above except without the third condition f(1R) = 1S. A rng homomorphism between (unital) rings need not be a ring homomorphism. The composition of two ring homomorphisms is a ring homomorphism. It follows that the class of all rings forms a category with ring homomorphisms as the morphisms (cf. the category of rings).In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism. (Wikipedia).

Homomorphisms in abstract algebra

In this video we add some more definition to our toolbox before we go any further in our study into group theory and abstract algebra. The definition at hand is the homomorphism. A homomorphism is a function that maps the elements for one group to another whilst maintaining their structu

From playlist Abstract algebra

RNT1.3. Ring Homomorphisms

Ring Theory: We define ring homomorphisms, ring isomorphisms, and kernels. These will be used to draw an analogue to the connections in group theory between group homomorphisms, normal subgroups, and quotient groups.

From playlist Abstract Algebra

Abstract Algebra | Ring homomorphisms

We give the definition of a ring homomorphism as well as some examples. http://www.michael-penn.net https://www.researchgate.net/profile/Michael_Penn5 http://www.randolphcollege.edu/mathematics/

From playlist Abstract Algebra

Ring Homomorphisms 2022

A lecture on Ring Homomorphisms.

From playlist Modern Algebra

Homomorphisms in abstract algebra examples

Yesterday we took a look at the definition of a homomorphism. In today's lecture I want to show you a couple of example of homomorphisms. One example gives us a group, but I take the time to prove that it is a group just to remind ourselves of the properties of a group. In this video th

From playlist Abstract algebra

Group Homomorphisms and the big Homomorphism Theorem

This project was created with Explain Everything™ Interactive Whiteboard for iPad.

From playlist Modern Algebra

Group Homomorphisms - Abstract Algebra

A group homomorphism is a function between two groups that identifies similarities between them. This essential tool in abstract algebra lets you find two groups which are identical (but may not appear to be), only similar, or completely different from one another. Homomorphisms will be

From playlist Abstract Algebra

Visual Group Theory, Lecture 7.3: Ring homomorphisms

Visual Group Theory, Lecture 7.3: Ring homomorphisms A ring homomorphism is a structure preserving map between rings, which means that f(x+y)=f(x)+f(y) and f(xy)=f(x)f(y) both must hold. The kernel is always a two-sided ideal. There are four isomorphism theorems for rings, which are compl

From playlist Visual Group Theory

Schemes 10: Morphisms of affine schemes

This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne. We try to define morphisms of schemes. The obvious definition as morphisms of ringed spaces fails as we show in an example. Instead we have to use the more su

From playlist Algebraic geometry II: Schemes

Isomorphisms in abstract algebra

In this video I take a look at an example of a homomorphism that is both onto and one-to-one, i.e both surjective and injection, which makes it a bijection. Such a homomorphism is termed an isomorphism. Through the example, I review the construction of Cayley's tables for integers mod 4

From playlist Abstract algebra

Lecture 28. Algebras

From playlist Abstract Algebra 2

Schemes 23: Valuations and separation

This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne.. We state a condition for morphisms of schemes to be separated in therms of discrete valuation rings, and apply this to the line with two origins and the proje

From playlist Algebraic geometry II: Schemes

Abstract Algebra | Properties and examples of ring homomorphisms.

We present some important properties of ring homomorphisms and give some examples. For instance we prove that 2Z and 3Z are isomorphic as groups but not rings. http://www.michael-penn.net https://www.researchgate.net/profile/Michael_Penn5 http://www.randolphcollege.edu/mathematics/

From playlist Abstract Algebra

Ring Homomorphisms: I an Ideal in R implies Φ(I) is an Ideal in S

Ring Homomorphisms and more Ring homomorphisms. In this video I'm talking about mapping and ideal in one ring makes an ideal in another ring. (if the map is onto) I hope you learn something, have a great day! Some of the links below are affiliate links. As an Amazon Associate I earn from

From playlist Abstract Algebra

Charles Rezk: Elliptic cohomology and elliptic curves (Part 2)

The lecture was held within the framework of the Felix Klein Lectures at Hausdorff Center for Mathematics on the 3. June 2015

From playlist HIM Lectures 2015

302.3A: Review of Homomorphisms

A visit to the homomorphism "zoo," including definitions of mono-, epi-, iso-, endo-, and automorphisms.

From playlist Modern Algebra - Chapter 17 (group homomorphisms)

Ring homomorphism

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