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I came across a theorem called Dorroh Extension Theorem while reading a textbook on ring theory. What the theorem essentially says is that any ring $R$ can be embedded in a ring $R^{\prime}$ with identity, i.e. there exists a subring $S^{\prime}$ of $R^{\prime}$ such that $R\cong S^{\prime}$. What I cannot understand is the following question.

Why does not this theorem simplify ring theory to the study of rings with identity?

The fastest answer that comes to my mind is that a subring of a ring is not necessarily a ring with identity. But is this the only reason? I'll be grateful for any help provided.

Edit: Proof of the Dorroh Extension theorem.

Consider $R\times\mathbb{Z}$. Define the operations as $(a,m)+(b,n)=(a+b,m+n)$ and $(a,m)(b,n)=(ab+an+mb, mn)$. Then $R\times \mathbb{Z}$ is a ring with identity $(0,1)$. And $R\times\{0\}$ is a subring of $R\times\mathbb{Z}$. Moreover $f:R\to R\times\{0\}$ given by $f(a)=(a,0)$ is an isomorphism. Hence the theorem "Any ring can be embedded in a ring with identity".

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  • $\begingroup$ Please give a reference where Dorroh's extension theorem is explicited. $\endgroup$
    – Jean Marie
    Commented May 21, 2017 at 4:56
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    $\begingroup$ @JeanMarie At this point, Janitha357 would presumably just be googling a resource which you can do just as well. The only thing Janitha357 can provide beyond that is verifying that it does correspond to what is in the book. Alternatively, you could ask Janitha357 to provide the exact statement of the theorem. $\endgroup$ Commented May 21, 2017 at 5:12
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    $\begingroup$ @JeanMarie I will edit my question so that you can see the exact statement of the theorem and it's proof. $\endgroup$
    – Janitha357
    Commented May 21, 2017 at 5:21
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    $\begingroup$ @Janitha357 : I think it's the same reason why Steinitz's theorem doesn't simplify the study of fields to algebraically closed fields. Indeed, any ring can be embedded in a unitary ring, but there are properties of non unitary rings that are special to them and that you can't see when looking at the bigger unitary field in which it is embedded. See this answer I gave that shows that some properties are drastically different in non unital rings : math.stackexchange.com/questions/2194671/… $\endgroup$ Commented May 21, 2017 at 10:08
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    $\begingroup$ Check this answer: math.stackexchange.com/a/16171/133781 $\endgroup$
    – Xam
    Commented May 21, 2017 at 17:32

2 Answers 2

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I have seen the embedding given as an excuse for not studying rngs more than once, but never a convincing justification.

For one thing, one would expect a robust demonstration of how properties of a rng are or are not preserved through the Dorroh extension. But that never appears, and in fact few people know such properties of the extension, and quite frequently properties are not preserved. Furthermore, examples of interesting rng theory exist, so it is not cut and dry as one is led to believe.

So the case for dismissing the study of rngs is not very substantive in print. It would be interesting to see someone implement and give a rigorous implementation of what can be remedied using Dorroh extensions, though.

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The well-known Dorroh adjunction of $1$ is not useful in many contexts because it doesn't preserve crucial properties of the source rng (e.g. regular, semiprime, artinian, Ore domain) and/or doesn't satisfy various minimality properties. Below is an alternative which addresses these issues.

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. $\ \ $ Excerpt:

enter image description here

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