How do we distinguish two different notations for cokernel and coimage?

[elias001]

@martinbn Isn't it supposed to be if i know basic commutative algebra, it would make studying either Algebraic number theory and Algebraic geometry easier.

 

[fresh_42]

Isn't it supposed to be if i know basic commutative algebra, it would make studying either Algebraic number theory and Algebraic geometry easier.

In my opinion, yes, but you don't need the five or nine lemmas for it. The rings $\mathbb{Z}_n$ are the standard examples in many cases, so it's necessary to understand them. Rings are a generalization of the integers, and modules are a generalization of vector spaces. So it helps to study those foundations before the general case, and if it were for having a reservoir of examples.

 

[fresh_42]

@fresh_42 @martinbn if we have $f(n+5\mathbb{Z})=n+5^2\mathbb{Z}$, then for $n=7$, then both $7+5\mathbb{Z}, 2+5\mathbb{Z}$ maps to $7+5^2\mathbb{Z}$. So you two are right.

The other way around: $7+5\mathbb{Z}, 2+5\mathbb{Z}$ are different elements in $\mathbb{Z}_{25}$ but identical in $\mathbb{Z}_5.$ So it maps the same element to two different targets, which is not allowed for a function. That means in technical terms "$f$ isn't well defined".

We only have a (surjective) homomorphism $\mathbb{Z}_{25} \longrightarrow \mathbb{Z}_5$ by mapping $n+25\mathbb{Z}$ to $(n\pmod{5})+5\mathbb{Z}.$

 

[elias001]

@fresh_42 in linear algebra, for matrices, one can calculate the equivalent of its kernel and cokernel. I have seen cokernel amd exact sequences being discussed in a few linear algebra texts. Some of those texts includes category theory. Also $\Bbb{Z}-$modules are not vector spaces. it seems i can't always rely on vector spaces as a crutch for examples.

 

[elias001]

@fresh_42 i think I got a bit confused when I thought about $np$ instead of $n$.

 

[fresh_42]

Also I have never encountered a module that is not a vector space.

I bet you have. Take any abelian group $\left(G,+\right).$ Then we can define
$$
z\cdot g=\begin{cases}
\underbrace{-g-g-\ldots -g}_{-z\text{ times }}&\text{ if }z<0\\
0&\text{ if }z=0\\
\underbrace{g+g+\ldots +g}_{z\text{ times }}&\text{ if }z>0
\end{cases}
$$
and $G$ becomes a natural $\mathbb{Z}$-module.

Also, every ideal in a ring is a module over that ring.

 

[elias001]

@fresh_42 i edited my post. I said with modules over the integers, I can't always rely on vector spaces as a crutch for sources of examples.

 

[martinbn]

@martinbn Isn't it supposed to be if i know basic commutative algebra, it would make studying either Algebraic number theory and Algebraic geometry easier.

Yes, but for an introductory course on either of those you don't need much to start with and a lot is usually covered in the text.

 

[elias001]

@martinbn I have never taken any introductory course in abstract algebra. I am at a point at Dummit and Foote where the examples being presented are from things that are beyond the introductory mark.

 

[martinbn]

@martinbn I have never taken any introductory course in abstract algebra. I am at a point at Dummit and Foote where the examples being presented are from things that are beyond the introductory mark.

You said that you are studying it on your own, not as a part of a course. My point is, in that case if you find it unmotivated study something else.

 

[elias001]

@martinbn I am also studying on my own so that I know how to do all the exercises in the first 14 chapters in Dummit and Foote since that will be the textbook my university will be using in their introductory abstract algebra course. Fourteen chapters of raw unadulterated abstract algebra served and forced fed to the students at an overdosed speed in the span of less than nine months. I have met students who fail that class. One professor said anyone she has seen that fell behind in that course never manages to catch up. In another occasion, she told me the material in Dummit and Foote was very basic as far as abstract algebra is concerned. I am not in a position to judge what she meant by being basic is suppose to mean.

 

[elias001]

@martinbn I think the two volumes of Samuel and Zariski were what Courant's two volumes of Differential and Integral calculus to the subject of Calculus and Analysis. Thee is no similar text for beginning level abstract algebra. The written style for both two-volumes were written in such s way the authors were giving a lecture to the reader in person.