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.