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It is well known that every non-unital ring can be embedded into a unital ring (e.g., Dorroh's adjunction). I am curious about the converse: every unital ring can be viewed as a subring of a non-unital ring?

If this converse is not correct, will this be still partially true? I am looking for a non-trivial example. One trivial example would be that $\{0\}$ as a subring of $2\mathbb{Z}$.


PS: by a ring here I mean a set that is an additive abelian group and a multiplicative semigroup, and satisfies the distributive laws.

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  • $\begingroup$ What do you mean by embedded? Do you mean that there is an injective ring homomorphism? Because any unital ring would have no homomorphism to a non-unital ring if morphisms must preserve $1.$ $\endgroup$ – Chickenmancer Aug 15 '18 at 4:49
  • $\begingroup$ @Chickenmancer You may have a look at here for what I mean:math.stackexchange.com/questions/1113097/… $\endgroup$ – Easy Aug 15 '18 at 5:10
  • $\begingroup$ @Chickenmancer I think we should note that a ring and its subring do not share the same "1" in general. $\endgroup$ – Easy Aug 15 '18 at 5:14
  • $\begingroup$ math.stackexchange.com/questions/170953/… They do share the same 1 in general. $\endgroup$ – Chickenmancer Aug 15 '18 at 6:32
  • $\begingroup$ @Chickenmancer Can you tell what are the "1"s in $\{0\}$ and $\mathbb{Z}$ respectively? $\endgroup$ – Easy Aug 15 '18 at 7:31
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Take any ring $S$ without identity. If $R$ is any ring with identity, then $R\times S$ does not have identity. Is this what you seek?

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  • $\begingroup$ Nice to see that you're gonna hit 100k reputation. $\endgroup$ – Xam Aug 16 '18 at 4:53

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