Notion of a ring with identity 1=0

[JohnKeats]

Could someone clarify the notion of a ring with identity element $1=0$? Apparently it's just the zero ring, but then why do we always talk about a ring with identity $1 \ne 0$? It's like having to talk about prime $p \ne 1$; instead we don't define $1$ as a prime but also because we would lose unique factorisation. So what would be lost if we said the zero ring is in fact not a ring?

 

[I like Serena]

Could someone clarify the notion of a ring with identity element $1=0$? Apparently it's just the zero ring, but then why do we always talk about a ring with identity $1 \ne 0$? It's like having to talk about prime $p \ne 1$; instead we don't define $1$ as a prime but also because we would lose unique factorisation. So what would be lost if we said the zero ring is in fact not a ring?

Hi JohnKeats, welcome to MHB! ;)

First off, I'm actually not aware that we 'always' talk about a ring with identity $1 \ne 0$.
Generally I think that for rings that is not really an important distinction.
Can you tell us where you have seen that?

Either way, rings are already a bit ambiguous, since different authors define them differently.

• Some authors do not include $1$ in a ring at all, making the point moot.
  This is why we see the phrase ring with identity (meaning multiplicative identity) to eliminate the ambiguity.
• Some authors include multiplicative commutativity in the definition, although that is not relevant now.

If we go a step up to a field, the distinction between $1$ and $0$ does become relevant.
In a field it is specifically required that $1 \ne 0$, so we don't need to mention it.
The reason is that a field without $0$ is supposed to be a multiplicative group.

 

[JohnKeats]

Hi JohnKeats, welcome to MHB! ;)

First off, I'm actually not aware that we 'always' talk about a ring with identity $1 \ne 0$.
Generally I think that for rings that is not really an important distinction.
Can you tell us where you have seen that?

Either way, rings are already a bit ambiguous, since different authors define them differently.

• Some authors do not include $1$ in a ring at all, making the point moot.
  This is why we do see the phrase ring with identity (meaning multiplicative identity) to eliminate the ambiguity.
• Some authors include multiplicative commutativity in the definition, although that is not relevant now.

If we go a step up to a field, the distinction between $1$ and $0$ does become relevant.
In a field it is specifically required that $1 \ne 0$, so we don't need to mention it.
The reason is that a field without $0$ is supposed to be a multiplicative group.

Thank you.

Indeed I was reading this in the context of fields. But also in the context of the definition of units. But these seem very related so it might be the same context.

 

[Olinguito]

Adding my humble opinion here. When rings were first studied, they were modeled on the integers; hence they had a multiplicative identity, and were also commutative. When Emmy Noether began studying noncommutativity and other deeper properties of rings, she aimed to be as general as possible and removed the requirement that rings should contain a multiplicative identity.

Wikipedia says:

Most or all books on algebra up to around 1960 followed Noether's convention of not requiring a 1. Starting in the 1960s, it became increasingly common to see books including the existence of 1 in the definition of ring, especially in advanced books by notable authors such as Artin, Atiyah and MacDonald, Bourbaki, Eisenbud, and Lang. But even today, there remain many books that do not require a 1.

Faced with this terminological ambiguity, some authors have tried to impose their views, while others have tried to adopt more precise terms.

In the first category, we find for instance Gardner and Wiegandt, who argue that if one requires all rings to have a 1, then some consequences include the lack of existence of infinite direct sums of rings, and the fact that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable."

Besides the technical objection voiced by Gardner and Wiegandt, there is an aesthetic appeal to not including $1$ in the definition of a ring: it means that all ideals are subrings. (In a ring $R$ with identity $1$, the only ideal containing $1$ is $R$ itself.) There is then a nice correspondence between ring theory and group theory with ideals and subrings being equivalent to normal subgroups and subgroups respectively.

I suppose the requirement of $1$ in the definition of a ring is that most of the ring-theory topics in textbooks (e.g. algebraic-number theory) require rings to have a multiplicative identity (and to be commutative); this is thus incorporated in the definition at the outset to avoid long-windedness (we can say “ring” instead of “(commutative) ring with multiplicative identity” every time). Moreover the requirement $1\ne0$ can be specifically incorporated in the definition to avoid the trivial case.

 

[I like Serena]

It looks to me as if we need a new entity to fill in the gap.
As it is, we have the hierarchy:

group ⊃ abelian group ⊃ rng ⊃ ring ⊃ commutative ring with distinct identity ⊃ integral domain ⊃ integrally closed domain ⊃ GCD domain ⊃ unique factorization domain ⊃ principal ideal domain ⊃ Euclidean domain ⊃ field​

As I see it, authors really shouldn't reuse a term that already has a specific meaning for their own purposes.
Instead they should coin a new term if that is needed.
And indeed, 'commutative ring with distinct identity' or 'non-zero ring with identity' are rather long winded.
We need a shorter name for it and anyone could invent one.
For instance cring, int-ring (integer-like ring), or commutative domain (my inventions).
Anyone has a better suggestion? ;)