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Page 31. Corollary 1.5. Let $k$ be a field, and let $S = k[x_1, \dots, x_r]$ be a polynomial graded by degree. Let $R$ be a $k$-subalgebra of $S$. If $R$ is a summand of $S$, in the sense that there is a map of $R$-modules $\varphi: S \to R$ that preserves degrees and takes each element of ...

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This is testing my understanding of algebras, subalgebras, & polynomial rings. Page 31. Corollary 1.5. Let $k$ be a field, and let $S = k[x_1, \dots, x_r]$ be a polynomial graded by degree. Let $R$ be a $k$-subalgebra of $S$. If $R$ is a summand of $S$, in the sense that there is a map of ...

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Here is the notation & motivation.: Conjecture: If $f \in \hat{m}$, then $f = \sum g_i f_i$ where each $g_i$ is homogeneous. I've tried proof by induction on degree of $f$, number of generators making up $f$, breaking up the $f_i$ into degree buckets. What am I not seeing?

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I think this is a very basic question but somehow I am unable to understand the answer. Find the number of zeroes of $f(x) = x^3 + x + 1$. Answer in the book : $f^\prime (x) = 3x^2 + 1$ since $3x^2 + 1 \ge 0$ for all $x \in \mathbb{R}$ therefore $f^\prime (x) \ge 1$ for all $x \in \m... 2 Find number of zeroes of$f(x) = 1 - x^{-2}$. I assume that this function has two or more zeroes in the domain$ \mathbb{R} - \{0\}$. Since$f^\prime (x) = \large{2\over x^3}$, therefore we can say$f^\prime(x) \ne 0$for all$x \in \mathbb{R} - \{0\}$. Therefore by Rolle's theor... 2 Suppose that, $$f(x)=\sum_{n=0}^{\infty}a_nx^n$$ converges on$(-R,R)$for some$R>0$. If we let$(x_n)$be a sequence in$(-R,R)$with$x_n\ne 0$but$limx_n=0$. If$f(x_n)=0$for all$n\in N$, I need to show that$f(x)=0$for all$x\in (-R,R)$. My thinking is that I could use Rolle's Theorem... 2 Is the property of local connectedness or local path-connectedness invariant under homotopy equivalence? I.e. If$X,Y$are homotopically equivalent Hausdorff topological spaces such that$X$is locally connected / locally path connected, then is it true that$Y\$ is also locally connected/locally ...

Keep? Remove? Synonymize?
There already is suggestion to make a synonym of .
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There are separate tag connectedness and path-connected. In the tag-info cor connectedness it is explicitly mentioned that it includes (among other thing) questions about path-connectedness. (Moreover, having too many closely related tags causes problem, since there is a limit at most 5 tags per ...

2 hours later…
1:27 PM
@MartinSleziak I wen through the posts tagged and removed the tag. So unless it is used again in the next 24 hours, the tag should be removed from the system.