I am currently studying Milnor $K$-theory and came across his definition of a transfer map which works if we have an inclusion of rings.
However, in Quillen's $K$-theory, he defines a transfer map for injections of rings.
My question is now: how is the norm/transfer map defined for an injection of rings? More precisely: if I have an injective ring homomorphism \begin{equation*} f:R\to S, \end{equation*}how is the transfer map \begin{equation*} f^*:K_2(B)\to K_2(A) \end{equation*}defined? I have read something about restriction of scalars but do not see how to apply it to a specific case if we're not talking about the category of all $R$- resp. $S$-modules.
Every reference I find just notes that is it 'well-known' and I cannot seem to construct it myself.
I'm happy about any suggestions and answers, thanks!