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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:

  1. Does Albert's adjunction have any name? Is it also called Dorroh's adjunction?
  2. Does the "best" way described in the aforementioned paper have a name?
  3. If we adjoin a $1$ to $R$ in the "best" way, is the extension ring $\kappa(R)$ or is it something else?
  4. Is there any equivalent formulation of the "best" way that is easier for a beginner to understand?
  5. In the "best" way, is the characteristic of the extension/adjoined ring different from the characteristic of $R$?
  6. 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?
Ramen Nii-chan
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  • The name is "Dorroh" not "Dorrah", throughout. Could you please correct that? – rschwieb Oct 17 '18 at 16:35
  • And it might be helpful to clarify that "doesn't preserve properties of $R$" refers to the properties not passing to the extension $R'$ (all properties remain intact in the isomorphic copy of $R$, of course.) – rschwieb Oct 17 '18 at 16:37
  • You may be interested in reading the review at MathSciNet, which is good: https://mathscinet.ams.org/mathscinet-getitem?mr=1018976 – Jose Brox Oct 17 '18 at 18:15

1 Answers1

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  1. 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.

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

  3. 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...

  4. 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.

  5. 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.

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

rschwieb
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