Why the definition of a ring does not include the identity element? I'm taking a course in abstract algebra 2, which basically about rings and fields. I noticed in several textbooks and lecture notes when they discuss a ring, and they say let R be a ring with Identity. My question why not including the definition of identity in the rings like groups? Also, are there interesting rings without identity? Like what?

please elaborate as much as you can I'm a novice in algebra.
 A: Whether a ring has an identity by definition is a matter of convention. Or, if you prefer, faith. I belong to the church teaching that rings must have an identity (and call the lesser structures rngs, they even have a tag here in Math.SE). This is because:

*

*I was raised that way (both as an undergrad as well as a graduate student). I use Jacobson's Basic Algebra I-II as my bible.

*In my career I have only worked with rings, where this is more convenient: modules (as with vector spaces) absolutely need the identity element to act by the identity function, fields (no contest) and rings in algebraic geometry/commutative algebra (again via modules).

There are other mathematicians who need rngs, and they choose to work with a different definition. As Kavi Rama Murthy explained in a comment, in analysis you run into situations, where you, for example, don't want to include the constant functions because they don't have compact support. Whatever, I trust they have a reason. Even though they could easily extend their rng by taking a direct sum with scalar multiples of $1$. But possibly they don't want to make allowances for that much the same way people working with modules don't want to add clauses stating that the results don't apply when the ring acts by all zeros. See also the comment by Jackozee Hakkiuz under the question.

Different needs by different mathematicians have left authors of algebra textbooks in a slightly uncomfortable place. Within the book you need to pick a definition and stick to it. Otherwise the students will be confused when meeting an exercise where the difference matters. The responsible authors mention the different conventions, because the students are likely to meet both. Or, at least some of the students reading the text will later encounter other texts, where different conventions are in place. The author's non-choice may only be glossed over, but in my opinion it would be irresponsible to pretend the alternative does not exist.


*

*If, in the remainder of the book, or lecture notes, you will never need/discuss rngs, it is more economical to define a ring to have a multiplicative neutral element. You save your readers a lot of headache that way.

*But, if you consult the entire department of colleagues then you, as the author of a set of lecture notes for a course designed to serve all the students, you should take their opinion into account, and explain the different conventions. Simply because both are widely used.



The question whether a ring has a $1$ or not is not unlike that of whether $0$ is a natural number or not. You can do math with either convention. The authors explain their conventions early in a text. And an educated reader knows to look for it (unless obvious from the context).

My creed. Basically God tells me that rings have $1$. But he also advised me that it is NOT my duty to convert the infidels.

Listing a few other reasons (that I can easily think of) in favor of including existence of $1$:

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*Certain constructions (like the field of fractions of an integral domain) need it. True, you can insist that an integral domain should always have $1$. Don't know how pressing this is.

*If you don't require $1$, then you run into scenarios where your ring actually has $1$, as does a subring, but the two "ones" need not coincide. This is a headache.

Listing a few other reasons (that I can easily think of) in favor of including rngs:

*

*The collection you get is more general.

*In some contexts it may be an advantage that ideals are automatically subrings. My experience is the opposite that it is an advantage that no non-trivial ideal can be a subring, but that is my upbringing talking.

A: Why wasn't the commutativity axiom included in the definition of the group?
Because commutativity is not something very obvious to be included in the definition of a group,I mean you can find a good number of structures where the operation isn't commutative,very good and important example is general linear group.
Likewise existence of  identity (here in rings identity for second operation) is not something very obvious to be included in the definition of a ring, because you find good number of structures where we are unable to find an element in the ring that acts as identity for second operation,as rightly pointed out in one of the comments 2Z is a very good example.
You have to remember that these definitions aren't something which were born in a day,it took years to develop,many Algebraic structures would have been studied and finally mathematicians,came up with the definition that we have now
A key point about these definitions is it has been formulated in such a way that it is simple enough to include good number of important examples,also it is not too simple enough to include all the examples
Also you might find books in which in the definition of ring itself they have included existence of identity,but many books don't also,it depends
