  Renewal processes are zero-dimensional processes defined by independent
intervals of time between zero crossings of a random walker. We subject renewal
processes them to stochastic resetting by setting the position of the random
walker to the origin at Poisson-distributed time with rate $r$. We introduce an
additional parameter, the probability $\beta$ of keeping the sign state of the
system at resetting time. Moreover, we introduce a trap at the origin, which
absorbs the process with a fixed probability at each zero crossing. We obtain
the mean lifetime of the process in closed form. For time intervals drawn from
a L\'evy stable distribution of parameter $\theta$, the mean lifetime is finite
for every positive value of the resetting rate, but goes to infinity when $r$
goes to zero. If the sign-keeping probability $\beta$ is higher than a critical
level $\beta_c(\theta)$ (and strictly lower than $1$), the mean lifetime
exhibits two extrema as a function of the resetting rate. Moreover, it goes to
zero as $r^{-1}$ when $r$ goes to infinity. On the other hand, there is a
single minimum if $\beta$ is set to one.
