I didn't understand why some books define subring in this manner: (Remark: this book is Steps in Commutative Algebra by Sharp)
Instead of this one:
Let R be a ring. Any intersection of subrings of R is again a subring of R. Therefore, if X is any subset of R, the intersection of all subrings of R containing X is a subring S of R. S is the smallest subring of R containing X. ("Smallest" means that if T is any other subring of R containing X, then S is contained in T.) S is said to be the subring of R generated by X. If S = R, we may say that the ring R is generated by X.
I know these definitions are equivalent, but I don't understand why the authors of these books don't define this in a more general way; why does he put the subring $S$ of $R$ in the former definition? We can simply define subring generated by a set as the intersection of the subrings which contains this set, period.
Is there some reason why these definitions are stated differently?
Thanks in advance.
The second definition treats the generators of the subring as a mere set, thereby ignoring any extra structure that may be present therein (for example part of the generators itself being a subring); this has the benefit of being universally applicable and lacks the arbitrary choice of "base" scalar ring upon which to build. The first definition, where $S[\Gamma]$ is the intersection of all rings above $S$ and containing $\Gamma$, emphasizes or distinguishes the generators in $S$ as different from that of $\Gamma$: this can be thought of as "adjoining" outside elements to $S$ to obtain $S[\Gamma]$. This coincides with notating of polynomial rings, as Chibchas states in the comments. Often the first definition is closer to our thought process in contexts where we use the concept of generated subrings.