Welcome to the forum.
I would think that PrE 10 would install and run on a Windows 8 computer, but do not know for sure. I just cannot think of a reason, with the OS, why it would not.
When you attempt to install the downloaded version of PrE 10, do you get any error messages? If so, what are they?
I assume that your version of Win8 is 64-bit. Is that correct? If so, which version of PrE 10 are you trying to install, 32-bit, or 64-bit?
If 64-bit, the 4GB of RAM will be a limiting factor, when running a program, like PrE 10, but should not affect the installation. You might want to look into expanding that RAM to at least 8GB, or better, more. A 64-bit OS requires a lot of resources, but can access them, and better.
This article is a checklist for the general installation of most Adobe products. You might want to look over it, in case you missed something, when trying to install: http://forums.adobe.com/message/4063309#4063309
One last thought: IIRC, PrE 10 download had two files: one was the program itself, with a separate installer, and the other file was the additional functional content, such as more Titles, Menu Sets, etc.. Some users mixed those two files up, and thought they were installing the program, when it was that additional functional content. That content installer would not work, until the program had been installed properly.
Good luck, and please let us know a bit more.
First, thank you very much for your response. I downloaded PrE 10 using the Aklai (?) download manager, and when it finsihed I had a folder labeled "Adobe Premiere Elements 10 Content". Inside are six folders and some other files of types I don't recognize. The folders are labeled Adobe Premiere Elements 10 Content 1 - 3, and Adobe Premiere Elements 10 HD Content 1-3. I assumed folders 1 & 2 were the 32 bit and 64 bit versions. However, when I open them, it says that there is not enough room to save them. I tried moving the downloaded content folder from my desktop to my C: drive program files, but that didn't work either. There is a setup and installer application that I can run inside the Content 1 & 2 folders, but a message appears that says that running either of them will mean removing the content I already downloaded. Am I looking in the wrong place?
Thanks for the "memory jog," in the change in the Additional Functional Content download. It is now a series of files, each with different Additional Functional Content. It WAS once just one file, with an installer package.
Those Content files are for more Title Templates, more DVD/BD Menu Sets, more Themes/Instant Movies, etc., and cannot be installed, until the PrE program, has been fully and successfully installed.