Reading Atiyah-MacDonald: Introduction to Commutative Algebra, I found the following definition of subring:
A subset $S$ of a ring $A$ is a subring of $A$ if $S$ is closed under addition and multiplication and contains the identity element of $A$. The identity mapping of S into A is then a ring homomorphism.
I know this definition is "wrong", as on the question I linked below is said:
Concept of a subring in Atiyah-Macdonald's book
But my question is, what if we change our "classic" definition of subring by this other? For me, it seems that everything remains equal and, at least, in the context of rings, there is not any contradiction.
For example, the following property is still true:
If $f: A \rightarrow B$ is a ring homomorphism, then $\operatorname{im}(f) \subset B$ is a subring.
I can't find any problem with this redefinition of subring. It will be welcome any correction or comment. Thanks everyone!
Best Answer
For clarity, let's use the following definitions (#-subring is not a standard term, just using it to distinguish the two possible definitions of subring).
Then any subring is a #-subring, so the property you've given clearly still holds. As will anything that proves some subset is a subring.
The problem is that a #-subring is not necessarily a ring: $\Bbb N$ is a #-subring of $\Bbb Z$, yet $\Bbb N$ is not a ring. Hopefully it is clear to you why a subring has to be a ring for it to be a sensible choice of definition!
A #-subring is, however, a semiring (a semiring has the same definition as a ring, but without the requirement for additive inverses - so in particular, any ring is a semiring), and in fact a #-subring of a ring $R$ is precisely a subsemiring of the ring (and thus semiring) $R$.