Adjoining $1$ to non-unital ring: is the usual way the best way? In Modern Higher Algebra written by A. Adrian Albert (1938), the characteristic of a non-unital ring $R$ is defined the least positive integer $m$ such that $ma=0$ for all $a\in\mathbb R$. If such an integer does not exist, the characteristic $m$ of $R$ is defined to be zero.
To adjoin a $1$ to $R$, Albert essentially embeds $R$ in a ring with additive structure $Z\oplus R$, where $Z=\mathbb Z$ when $m=0$ and $Z=\mathbb Z/m\mathbb Z$ when $m>0$. Ring multiplication is then defined as $(z,r)\cdot(z',r')=(zz', rr'+zr'+z'r)$, where $zr'$ is understood as the sum of $z$ copies of $r'$.
Note that Albert's adjunction ties the choice of $Z$ to $m$. When $Z$ is always taken to be $\mathbb Z$ regardless of $m$, the adjunction is commonly known as Dorroh's adjunction.
To my understanding, Albert's adjunction (rather than Dorroh's) is the "usual" way to adjoin $1$ to $R$.
It was mentioned on this site that Dorroh's adjunction is not very useful because it does not preserve many crucial properties of $R$, and in the following paper, there is a "best" way to adjoin a $1$ to $R$ that makes use of something called a characteristic ring $\kappa(R)$:

W.D. Burgess; P.N. Stewart. The characteristic ring and the "best" way to adjoin a one. J. Austral. Math. Soc. 47 (1989) 483-496.

As I have only studied abstract algebra at undergraduate level, I have trouble understanding the details of the paper. Here I am only looking for some some quick answers for the questions below:


*

*Does Albert's adjunction have any name? Is it also called Dorroh's adjunction?

*Does the "best" way described in the aforementioned paper have a name?

*If we adjoin a $1$ to $R$ in the "best" way, is the extension ring $\kappa(R)$ or is it something else?

*Is there any equivalent formulation of the "best" way that is easier for a beginner to  understand?

*In the "best" way, is the characteristic of the extension/adjoined ring different from the characteristic of $R$?

*In the "best" way, when the extension ring has the same characteristic as $R$'s, am I correct to say that it must contain Albert's extension ring as a subring?

 A: *

*I think I've seen in some contexts the construction using Dorroh multiplication on $R\times S$ where $S$ is an $R,R$ bimodule called "the Dorroh extension" too, although perhaps most people use it to refer to the choice of $R=\mathbb Z$.  So it may or may not be more general depending where you read it.

*If there were a good name for it, it would appear in the paper you're referencing.

*No, $\kappa(R)$ is a central subring of $R$ that is like the "prime ring" of a ring or "prime field" of a division ring, although it is not exactly those.  I think perhaps you need to read section 1 a little more closely...

*From definition 2.2, $R^1$ is a subring of $End(R_R)$, whose characteristic obviously matches that of $R$ (not a hard exercise.) THis only applies to left-faithful rings of course, so that $R^1$ is defined there.

*You've seen that it obviously can be different: the classical Dorroh extension with $\mathbb Z$ always has characteristic $0$. It is entirely dependent on the adjunction you choose.

*Yes. The Dorroh construction, even with $\mathbb Z/n\mathbb Z$, contains an isomorphic copy of $R$ (the set $\{0\}\times R$.)
