Recent content by Bashyboy

Homework Statement A nonzero free abelian group has a subgroup of index ##n## for every positive integer ##n## Homework EquationsThe Attempt at a Solution If ##F## is a nonzero free abelian group, then ##F## is isomorphic to the direct sum ##G= \sum_{i \in I} \Bbb{Z}##, where ##I \neq...

Sorry, I forgot to mention to stipulate that ##n \ge 2##.

Homework Statement Consider ##F[x,y]##, where ##F## is some field. I've been working on a problem all day and I'm having trouble with this last step. I am trying to show that ##x \notin(x,y)^n## for any ##n \in \Bbb{N}##. Homework EquationsThe Attempt at a Solution Note that ##(x,y)^n =...

Sorry. It should read "Prove that ##\mathcal{K}##..." I just edited it. Besides that, I typed up the problem word for word.

Homework Statement Assume that ##H## is a normal subgroup of ##G##, ##\mathcal{K}## is a conjugacy class of ##G## contained in ##H##, and ##x \in \mathcal{K}##. Prove that ##\mathcal{K}## is a union of ##k## conjugacy classes of equal size in ##H##, where ##k = |G : HC_G(x)|## Homework...

https://math.stackexchange.com/questions/2578120/ring-of-quotients-and-the-radical-of-an-ideal

Okay. Here is another try. If ##x \in \mbox{ Rad }(S^{-1})##, then ##x^n \in S^{-1}I## for some ##n \in \Bbb{N}## and therefore ##x^n = \frac{r}{s}## with ##r \in I##. But ##x## is also an element in ##S^{-1}R##, so that ##x = \frac{t}{k}## for ##t \in R## and ##k \in S##. This implies...

Well, ##I \subseteq R##, so ##S^{-1} I \subseteq S^{-1} R##.

Let me try again. Suppose that ##x \in \mbox{Rad}(S^{-1}I)##. Then ##x = \frac{r}{s}## with ##s \in S## and ##r \in R## such that ##x^n = \frac{r^n}{s^n} \in S^{-1} I##. Then by definition ##r^n \in I## and ##s^n \in S## and therefore ##r \in \mbox{Rad } I##. Hence ##\frac{r}{s} \in S^{-1}...

Homework Statement Let ##S## be a multiplicative subset of a commutative ring ##R## with identity. If ##I## is an ideal in ##R##, then ##S^{-1}(\mbox{ Rad } I) = \mbox{Rad}(S^{-1}I)##. Homework EquationsThe Attempt at a Solution If ##x \in S^{-1}(\mbox{ Rad } I)##, then ##x = \frac{r}{s}##...

Yes. At those values, ##f## will take on an extended real value (i.e., ##\pm \infty##). Then just subtract them, which I will explain below. Hmmm...It's possible. I took the problem statement mean that I am to prove the following: Assume that ##m(E)< \infty##. Let ##f_n : E \to \Bbb{R}##...

Homework Statement Show that Egoroff's theorem continues to hold if the convergence is pointwise a.e. and ##f## is finite a.e. Homework Equations Here is the statement of Egoroff's theorem: Assume that ##E## has finite measure. Let ##\{f_n\}## be a sequence of measurable functions on ##E##...

Ah! I see. What book are you referencing? By the way, given that that result has been proven, does my proof seem right?

Well. I didn't include that because I figured that it is a standard result.

Yes, I have proved this already.