This should be a remarkably easy question, but one which I am curiously finding myself unable to find a formal definition of on the internets: is a tower of rings simply defined in the same way as that a tower of fields, i.e., if $R$ is a subring of $S$, which in turn is a subring of $T$, then $R \subseteq S \subseteq T$ is a tower of rings?
The reason why I am asking this question is because Weibel in his book on $K$-theory at one point mentions "the tower of rings $R/I^n$", but, well, it is not necessarily the case that given an ideal $I$ of a ring $R$ that $R/I$ is isomorphic to a subring of $R$. Further, while one can of course construct surjections such that we have a sequence $$R \twoheadrightarrow R/I \twoheadrightarrow R/I^2 \twoheadrightarrow R/I^3 \twoheadrightarrow R/I^4 \twoheadrightarrow \dots \twoheadrightarrow R/I^k \twoheadrightarrow R/I^{k+1} \twoheadrightarrow \dots$$ those are surjections, not injections. I furthermore on Wikipedia learn that the formal definition of a ring extension appears to be a bit more sophisticated than that of a mere field extension. Which makes me wonder, maybe the formal definition of a power of rings has something to do with how a long exact sequence can be thought of as a succession of short exact sequences.
Would very much appreciate to have this sorted out, thank you.