I have submitted my website to Google Webmaster Tools with both
www and with
no-www. I had made the domain without
www to my preffered domain and have submitted a sitemap.
My question is should I also submit a sitemap to the one whit
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You should submit the sitemap only to the version of the site that matches the URLs in the sitemap.
Submit this sitemap only to the no-www site in Google Webmaster Tools:
Submit this sitemap only to the www site in Google Webmaster Tools:
The URLs in your sitemap should match your preferred domain preference. Google uses which URLs appear in a sitemap as one way of figuring out canonical URLs (see The Sitemap Paradox). Your preference setting in webmaster tools is another way. In order not to confuse Google, the two should match.
So in your case, I would expect all the URLs in your sitemap to include the
www. and you to submit the sitemap to the www site in Google Webmaster Tools.