I have strange issue, word document open in browser as expected. But a read permission user open up the document in browser and able to edit and save it.
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Have you checked the permissions on the document level? A user having read permission on the document library level might have write permissions set on the folder or document level. Check the permissions set for group and the group memberships (like AD users and groups).
The user might have extra permissions, like:
Site collection administrator
User policy (set in Central Administration for the web application) may allow users to make changes even without without explicit permissions on items