Let $A$ be a ring such that for any $x \in A, x^2=0$ implies $x=0$.  Show that if $a \in A$ satisfies $a^3=a$ then $2a$ is in the center of $A$.  I have no idea how to use the $x^2=0$ implies $x=0$ thing. My progress is as follows: note that if $a^3=a$ then $6a = 0$.  Denote by $B$ the set of $a$'s in $A$ for which $a^3=a$. I suspect that $B$ actually induces a subring. First of all, $B$ is closed under taking inverses: if $a^3=a$ then $(-a)^3 = - (a^3) = -a$ so $-a \in A$. Also $B$ is closed under addition: if $a \in B$ then $(2a)^3 = 8a^3 = 8a = 2a; (3a)^3 = 27a^3 = 3a$ and $4a = - 2a, 5a = -a$.   Let's say this actually induces a subring. Then the subring is of characteristic at most $6$, so it is of characteristic $1,2,3$ or $6$. If it is of characteristic $1$ or $2$ it is trivial that $2B$ is in the center of $A$. Otherwise I dont know how to proceed.  Edit: indeed I don't think this forms a subring. I proved it: here is a quick outline: Since the ring is reduced, any idempotent element is central. If $a=a^3$ then $a^2=a^4$ so $a^2$ is idempotent and thus central. We can show that if $a^3=a$ then $(a+1)^2$ is idempotent and thus central. Then since the center of the ring is a subring, we have that $(a+1)^2-a^2-1$ is in the center so $2a$ is in the center.   The problem now asks to show that any two elements of $B$ commute; That is, if $a^3=a$ and $b^3=b$ then $ab=ba$. Haven't cracked this one yet so ideas are welcome.
Let $(A,+,\cdot)$ be a unity ring with the property that if $x \in A$ and $x^2=0$ then $x=0$. Consider the set $M=\{a\in A | a^3=a\}$. Prove that:      a) $2a\in Z(A)$, $\forall a\in M$, where $Z(A)$ denotes the centre of the ring $A$;   b) $ab=ba$, $\forall a,b\in M$.   My attempts revolved around the fact that an idempotent element in a reduced ring is central. So, since for $a\in M$ we have that $(a^2)^2=a^2$, it follows that $a^2\in Z(A)$, $\forall a\in M$. The next thing I wanted to use in order to solve a) was that $Z(A)$ is a subring of $A$, so if I had proved that $(a+1)^2 \in Z(A)$, $\forall a\in M$, then we would have reached the desired conclusion. However, I couldn't prove this and I honestly doubt that it is true. Another idea that I had was to prove that $M$ is a subring of $A$. Of course, this didn't work out because I cannot even prove that $M$ is closed under addition. Again, I don't know if this is true and it most likely isn't. As for b), I think that a) should be of use, but I don't know how. It is a well-known problem that a ring with $x^3=x$ for any $x$ in that ring is commutative, but since $(M,+,\cdot)$ is almost definitely not a ring, this doesn't help. EDIT: Is there any chance that this question is simply wrong? I tended to believe this before asking it here too, but since nobody has made any progress on it until now I am even more inclined to think so.
How To Make Clickable Text In Minecraft I am trying to make clickable text in Minecraft in the chat box. It was used in the adventure map, "UP".   The idea is that you are given an option in the text to click, "Yes", or "No". Each word will give an output of a command.  How is this achieved? Which is the correct tense to use with 'by'? I gave him the money by 6 PM.   I had given him the money by 6 PM.   I can't spot the difference between them and both seem right, but I was told the first one is wrong. Can anyone say, why?  Here is a another set of sentences with the same issue:     By 5 PM I gave him the book.   By 5 PM I had given him the book.   In both the sentences the book was given before 5 PM and no later than 5 PM. But I can't understand then what the difference between them is.   Is it right to use simple past with 'by' at all? If yes, then why and if not, then why is it incorrect to use simple past with 'by'? 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In a Cycles scene I have an equirectangular image serving as an Environment Texture background (set in the World settings).  I have a glossy object being partially lit by the Environment Texture, as well as getting reflections from it.  Is there a way I can exclude the Environment Texture from affecting the glossy object completely? I want the image as background wallpaper only. The glossy object should be affected by other objects and lights within the scene: everything except the Environment Texture. Passing custom string into a Salesforce email template. Find minimum value of $\sum \frac{\sqrt{a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}$ I am doing an analysis with 2 groups of patients (control and disease) whose blood pressures have been measured continuously for 24 hours. My first goal is to test the effect of GROUP on BP. In the beginning, I built the mixed effect model with TIME as random effect variable and group as fixed effect model. 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I'll be 60 on my next birthday and I know that many Ph.D programs are traditionally geared toward younger students.I was wondering are there any programs that I can continue this research in without facing the issues of ageism in academia? How to copy a file to multiple directories using the gnu cp command Why would spacetime curvature cause gravity? What does the Web.Config file do in the views folder of a MVC project Entries I can safely exclude doing backups How can I change what keys on my keyboard do? (How can I create custom keyboard commands/shortcuts?) Using a comma before "but" My model has a clear plastic (or glass) dome with objects inside.  The objects appear dark.  I've tried the glass shader with color 100% RGB, with very small IOR, and using just a refraction shader.    I know Cycles has problems with caustics, but I just want broad illumination to fall clean through the glass.  The light-to-column path can ignore refraction, but is blocked or tremendously attenuated. 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