Why is it necessary for a ring to have multiplicative identity?
I have read earlier that in a ring $(R,+,.)$ the following needs to hold:
- $(R,+)$ is an abelian group
- multiplication is associative and closed
- left and right distribution laws hold.
However, I recently came across the fact that every ring has to have a multiplicative identity. Can anyone please clarify this? Is it needed for the ring to have a multiplicative identity?
(In fact it was mentioned that it is one of the reasons why $ker(f)$ is not a subring where $f$ is a ring homomorphism as the additive identity and the multiplicative identity are not usually in the same subset.)
Further in 2 different places I have noticed that there is a difference on whether the mapping $f(1) \to 1$ is a necessary condition for $f$ to be a ring homomorphism. I think this is also related to my doubt as to whether the multiplicative identity is in fact a necessary condition for defining a ring.
Many authors take the existence of $1$ as part of the definition of a ring. In fact, I would disagree with Alessandro's comment and claim that most authors take the existence of $1$ to be part of the definition of a ring. There is another object, often called a rng (pronounced "rung"), which is defined by taking all the axioms that define a ring except you don't require there to be a $1$.
Rng's are useful in of themselves, for example functions with compact support over a non-compact space do not form a ring, they form a rng. But there is also a theorem that states that every rng is isomorphic to an ideal in some ring. So studying rings and their ideals is sufficient, and this is why it is so popular to include the existence of $1$ as one of the axioms of a ring.
So to summarize, there isn't really a reason why it's necessary for rings to have a $1$, it certainly does not follow from the other axioms. It's just a choice of terminology: Do you say rings have a $1$ and if they don't have a $1$ call them rngs, or do you say rings don't need a $1$ and when they do have it call them rings with unity?
I'm currently teaching out of the 4th edition of Stewart's Galois Theory textbook. Stewart defines a ring to be what other authors might call a commutative ring with unity. The reason is simple: in this book, there is not much call for noncommutative rings, nor for rings without unity, and it gets old writing "commutative ring with unity" over and over, when that's the only kind of ring you need.
Stewart then defines a subring of a ring to be a subset of a ring closed under addition, subtraction, and multiplication. Note that a subring doesn't have to have unity – a subring doesn't have to be a ring, in this book. Well, it's a convention. As long as it's explained to the reader, and the author is consistent with it, I think it's fine.
Then he goes and spoils it by asking, in Exercise 16.2, whether the rings $\bf Z$ and $2\bf Z$ are isomorphic.
Lang and a few other authors use "Ring" to mean "Ring with unity" and say "Ring without unity" for what I'd call a Ring.
This is because Rings with unity are by far the most interesting. There are few things you can say of/do to a ring (or ring without unity to you) but there are MANY MANY things you can do with rings with unity (rings to you)