It’s called Dorroh’s extension? I never knew that (I know the construction of course; I just never heard the name). I can’t think of much exciting to say about it. Let me note that the formal expression is meant to be intuitively thought of as , and that the rules for addition and multiplication follow this intuition straightforwardly. Also, the embedding is given by .

I will say that while some authors’ convention is that a ring need not have a unit, that seems to be a bit old-fashioned: more and more, the default seems to be that rings have units by definition, and that “rngs” are what we call the structures with no unit assumption. (The term is due to Jacobson; think “ring” without the “i” [the identity].) I think the old-fashioned convention was partly there so that ideals of rings, or kernels of ring homomorphisms, would also be rings, but it’s perhaps more useful nowadays to think of ideals of a ring instead as submodules of (considered as a module over the ring ).

I’m not sure how compelling an argument I can give for why rings “ought to” have units (why that is the better convention), but generally units are extremely convenient for many arguments. The notion of category includes units (identities) for good reason: it makes possible the notion of isomorphism, the argument of the Yoneda lemma, and many other things. Not having units is an inconvenience, and this is just as true for monoids and for rings. Dorroh’s extension shows that we can harmlessly adjoin a unit anyway [and also that every rng arises as an ideal of a ring], so why not assume it’s there? So there are good arguments I think for Jacobson’s convention.

One naturally occurring example of a rng (i.e., where a unit is not naturally available) is the algebra of under convolution product; a unit would be a Dirac functional (a distribution not given by an function). But usually units are “naturally” part of the picture.

Notice that even if already has a unit, this construction adjoins a new unit. This is in contrast to other situations: for example, considered up to isomorphism, the field of fractions construction (for an integral domain) doesn’t adjoin extra elements if the domain was already a field. The fact that Dorroh’s extension adds something new even if the rng is a ring is connected with the fact that a rng homomorphism between rings need not be a ring homomorphism, because a rng homomorphism need not preserve identities.

]]>On a somewhat different note, I was mulling over a construction called *Dorroh’s extension*, which says that any arbitrary non-unital (or unital) ring can be embedded in a unital ring. Indeed, let be our arbitrary ring, then , together with and , is the aforesaid extension (with multiplicative identity as in which is the additive identity of ) and with being the required embedding. I was wondering if there is anything special about the above construction.

You used distributivity on both sides to prove your result. Here’s an example which shows you had to do that. Let be a group, with group multiplication denoted by . Define a new operation by . Then we have one-sided distributivity

and the identity element of is also a left identity for , but obviously this doesn’t force commutativity!

]]>As an aside, it is interesting to note the following: Commutativity of addition in a ring follows if the ring has identity . (Of course, commutativity of addition is already implied from the definition of a ring.) Anyway, using the two distributive laws, we note that , and that the same expression is also equal to . And so, through additive cancellation, we have for all in .

]]>Michael Rios says that these correspond to the 27 functions on 0,1,2 which he associates with the 3×3 octonion Jordan algebra (which is physically important). As for the physics, there are plenty of links on my blog. I’m afraid proper references are hard to come by.

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