  With recent discoveries of superconductivity in infinite--layer nickelates,
and in La$_{3}$Ni$_{2}$O$_{7}$ under high pressure, new opportunities appeared
that yet another family of high--temperature superconductors based on Ni
element may exist in Nature as was previously the case of cuprates and iron
based materials. With their famous strong Coulomb correlations among 3d
electrons and the proximity to antiferromagnetic instability these systems
represent a challenge for their theoretical description, and most previous
studies of superconductivity relied on the solutions of simplified few--orbital
model Hamiltonians. Here, on the other hand, we use a recently developed
combination of density functional theory with momentum and frequency resolved
self--energies deduced from the so--called Fluctuational--Exchange (FLEX)--type
Random Phase Approximation (RPA) to study spin fluctuation mediated pairing
tendencies in La$_{3}$Ni$_{2}$O$_{7}$ under pressure. This methodology uses
first--principle electronic structures of an actual material and is free of
tight--binding parametrizations employed in model Hamiltonian approach. Based
on our numerical diagonalization of the BCS Gap equation we show that competing
d$_{xy}$ and s$_{\pm }$ pairing symmetries emerge in superconducting
La$_{3}$Ni$_{2}$O$% _{7}$ with the corresponding coupling constants becoming
large in the proximity of spin density wave instability. The results presented
here are discussed in light of numerous other calculations and provide
on--going experimental efforts with predictions that will allow further tests
of our understanding of unconventional superconductors.
