Given a Ricci tensor $$R_{kk} = -(g_{kk})/a^2 - \delta_{ki}\delta_{ik}g_{kk}/a^2 + \delta_{jk}\delta_{ik}g_{kk}/a^2 + \delta_{ki}\delta_{ik}g_{kk}/a^2.$$
What's its correct simplified form? Here's my trial: $$a^2R_{kk} = -g_{kk} - n g_{kk} + \delta_{ij}g_{kk} + ng_{kk}\\ = -g_{kk} +\delta_{ij}g_{kk}.$$
From the result above, I see that $R_{kk} = 0$ for $i=j$ and $R_{kk} = -g_{kk}/a^2$ for $i\neq j$. Should I assume that, when $i=j$, $\delta_{ij} = n$, where $n$ is the dimension of the manifold?
I know it may look trivial for some of you, but I'm starting into General relativity and these manipulations with tensors and the einstein sum convention is still confusing to me.
Thanks in advance for any help.