  Let $p$ be an odd prime, $ f$ be a $ p $-ordinary newform of weight $ k $ and
$ h $ be a normalized cuspidal $ p $-ordinary Hecke eigenform of weight $ l <
k$. In this article, we study the $p$-adic $ L $-function and $ p^{\infty}
$-Selmer group of the Rankin-Selberg product of $f$ and $h$ under the
assumption that $ p $ is an Eisenstein prime for $ h $ i.e. the residual Galois
representation of $ h $ at $ p $ is reducible. We show that the $ p $-adic $ L
$-function and the characteristic ideal of the $p^\infty$-Selmer group of the
Rankin-Selberg product of $f, h$ generate the same ideal modulo $ p $ in the
Iwasawa algebra i.e. the Rankin-Selberg Iwasawa main conjecture for $f \otimes
h$ holds mod $p$. As an application to our results, we explicitly describe a
few examples where the above congruence holds.

Number Theory