Is an absorbing state necessarily recurrent? 
Definition:
For a state i, state i is an absorbing state IFF the probability that state i returns to state i, $p_{ii}$, is 1 and $p_{ij}=0$
Definition:
A state i is recurrent/ persistent if the probability of state i returning to state i k-times is $p^{k}_{ii}=1$

From here, it seems that an absorbing state is a recurrent state.
Am I correct with my deduction?

You are correct: an absorbing state must be recurrent.
To be precise with definitions: given a state space $X$ and a Markov chain with transition matrix $P$ defined on $X$.
A state $x \in X$ is absorbing if $P_{xx} = 1$; neccessarily this implies that $P_{xy} = 0, \, y \neq x$.
Given $x \in X$, the first return time is the random time at which the Markov chain first revisits $x$,
 $$\tau_x^+ = \inf\{ n \geq 1 \, \colon \, X_n = x\}.$$
A state $x \in X$ is recurrent if $\mathbf{P}_x[ \tau_x^+ < \infty] = 1$; that is if the Markov chain almost surely revisits $x$ having started at $x$.
Now if $x$ is an absorbing state, then we have
$$ \mathbf P_x[ X_1 = x] = p_{xx} = 1,$$
which is to say  $\mathbf P_x[ \tau_x^+ = 1] = 1$, and in particular we have $\mathbf P_x[\tau_x^+ < \infty] = 1$, which is to say $x$ is recurrent.
