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 Nov6 comment Book about Tensor Product of Vector Spaces Well... I guess you did not understand my question or I was not clear enough. I am not saying that I do not know this stuff. I am just asking for a reference! Oct27 comment Cardinality of a vector space versus the cardinality of its basis I guess I got it, but to be sure that I correctly understood: what you mean by $(F\times B)^{<\omega}$? Mar22 comment Tilting modules Let me try to clarify something: in the setting which I am talking about, a tilting module is a module with a filtration by standard and costandard modules. The category has the property that any tilting module is isomorphic to a direct sum of indecomposable tilting modules. I am not sure about this name "characteristic tilting module". Mar21 comment Tilting modules @Mariano: Thanks for the answers! I tried to make the question 2) more clear. What textbook do you suggest me? Feb15 comment Does the Implicit mapping theorem imply the inverse mapping theorem? @copper.hat I know about the equivalence, but I realized that I dont know how to prove it. With your suggestion, I found an application $C^1$ such that $f(g(x))=x$ under some conditions. I dont see why $g$ is the inverse of $f$, i.e $g(f(x))=x$. How should we proceed with your suggestion? Dec27 comment Dimension of a certain quotient ring of $\mathbb{C}[x_0,\ldots,x_{m-1}]$. Thanks for the suggestion by using MAGMA. I wrote a procedure to calculate the Grobner basis for $I$ and the linear basis for $A/I$ and I am obtaining the answer until $m=9$ too. I will leave open the question to see if anyone can suggest anything else. Dec27 comment Dimension of a certain quotient ring of $\mathbb{C}[x_0,\ldots,x_{m-1}]$. @Alexander Gruber: I am sorry, I forgot to lift the power of $u$ in the definition of $X(u)$. As you have spent your time doing calculations, I added in the question the example of the case $m=2$. Thanks. Nov23 comment Applications of the Jordan-Hölder Theorem. No way. I will not delete the post. As you said, even with a bounty, we don't have any other answer and your answer is the best which you can do! Thanks! Nov23 comment Applications of the Jordan-Hölder Theorem. However, the answer is pretty good in a general overview and it will be great if a person like you could also say something related to groups. Moreover, I respect the decision of "spin" who understood my question. Thanks! Nov23 comment Applications of the Jordan-Hölder Theorem. I am the same user Matt S and I am repairing it with stackexchange.com team. The original question has a context very clear and it is about groups, it is starting talking about textbooks on groups. I really understand you and respect your dedication in answer. As you said, this theorem is useful in many contexts, not explicitely where we expect, and this was the motivation of my question! I localized the subject saying about a book on group theory and its usual application, I disagree that to talk about its version for modules is giving an answer here! (continue) Nov21 comment Applications of the Jordan-Hölder Theorem. @spin: I respect everything described above and I agree with many parts which I understood. However, there is something much more subtle here: the title of the question is general, but the question is devoted for groups. So, the point is that the answer above is talking about extension of this theorem to the context of modules and not keeping the attention to groups. The samething could be done talking about its extension for suitable abelian categories, but it also means that the realm of groups is being left! Nov20 comment Wedderburn-Artin theorem Patrick: Thanks. I have some good level in mathematics and I have never heard this second part which I mentioned. The cyclic part is of course OK. But thanks in clarify me the second one! Nov20 comment Wedderburn-Artin theorem The module $B$ is a typo, right? You mean $A$ I guess. Apr19 comment Surjective homomorphism on Laurent polynomial ring, part II Hi, could you give a look at my other question in link. Thanks, Jan22 comment PBW Theorem applied to graded Lie algebras @user8268 The gradation is in $\mathbb Z_+^n$, then something more is necessary. There is no meaning for $b_1<\cdots< b_m$ in this case. Do you know where there exists a proof in the case of $\mathbb Z_+$-gradation? I think that it is possible to extend it... Otherwise, you could write one here to help! Jan22 comment PBW Theorem applied to graded Lie algebras @user8268: I think that he wants a decomposition of each piece in terms of tensor products of symmetric powers of ${\frak a}[r_i]$ for suitable choice of $a[r_i]$. I don't know how to do either. Dec16 comment Surjective homomorphism in Laurent polynomial ring. Thank you for helping me to do in the better way! I was not realizing the importance of this hypothesis which was inserted in the new question link. It was very helpful to understand what happens if I take out this hypothesis on my original problem. See the other question if you can! Dec16 comment Surjective homomorphism in Laurent polynomial ring. Good point! You gave me the full answer of a part of my problem. Actually, the problem which I am working has a stronger hypothesis that $a_i^k \ne a_j^k$ for $i\ne j$ and $k=1,2,3$. So, the counterexample that you gave is of the form $(t-1)(t+1)$ and therefore it is not satisfying my original hypothesis. I am very greatful about your answer and I am editing the original question with this variation. Dec15 comment Surjective homomorphism in Laurent polynomial ring. @jspecter: You are right! Dec15 comment Surjective homomorphism in Laurent polynomial ring. @Bill: I just put $a_i\ne 0$, that is a different question. Sorry about my failure!