Statistical test interpretation


Firstly, sorry if my english is bad, I'm French.

I wondered if I understood well how to interpret the result of a test, such as the Student test for the comparison of two means.

If I'm not mistaken, and from what I have read and understood the comparison of the p-value at alpha risk and value t confidence intervals are two different historical approaches to for interpretation of statistical tests.
Either we:
* Compare the p-value at alpha risk: if p-value> alpha, then we do not reject the null hypothesis H0, otherwise we reject it.
* Look if t belongs to the confidence interval (which is the complement of the critical region). If t belongs to the critical region, so we reject H0.

Therefore a method is sufficient to reject the null hypothesis or not.

Am I right?

Thank you in advance,