Let $R$ be a ring and $S$ be a subring of $R$. Let $\lambda_1,...,\lambda_n \in R$. I encountered $S[\lambda_1,...,\lambda_n]$, which is the smallest subring to contain both $S$ and $\{\lambda_1,...\lambda_n\}$.
I tried to visualise this but I find this quite hard to do. I thought this could be expressed as $$S':=\{r_0+r_1\lambda_1+...+r_n\lambda_n:r_0,...r_n\in S\}.$$ However I quickly realise this is very incomplete as $S'$ does not necessarily have to contain, $\lambda_1 \lambda_2$, say, which is obviously contained in $S[\lambda_1,...,\lambda_n]$. Hence could anyone help me to visualise $S[\lambda_1,...,\lambda_n]$?
It can be viewed simply as the image of the map $$S[x_1,\ldots,x_n]\ \longrightarrow\ R:\ x_i\ \longmapsto\ \lambda_i.$$ That is to say, every element of $S[\lambda_1,\ldots,\lambda_n]$ is a polynomial expression in the elements $\lambda_1,\ldots,\lambda_n$ with coefficients in $S$.