Does the Dorroh Extension Theorem simplify ring theory to the study of rings with identity?

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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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.