Your question is about the difference between past simple and present perfect.
"I checked my mail" -- this is past simple, formed by taking the past tense form of the verb "check". It simply means that at some time in the past you checked your mail.
"I've checked my mail" -- this is present perfect, formed by taking the verb "have" and adding to it the past participle of the verb "check". We use this tense when talking about a connection between the past and the present. In this case, you checked your mail in the past, but the connection with the present is: Now you know what is in your mailbox.
Sometimes it's hard to know whether you should use present perfect or past simple. But if you mention a time in the past when the action took place, you are only talking about the past -- you are not talking about any connection with the present.
So that means that if we say something happened on a particular day or at a certain time or in a particular year in the past, we must use past simple.
"3 minutes ago" is a time in the past. You checked your mail 3 minutes ago.
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