4

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.

Easy
  • 4,485
  • 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 $1.$ – Chickenmancer Aug 15 '18 at 04:49
  • @Chickenmancer You may have a look at here for what I mean:https://math.stackexchange.com/questions/1113097/is-it-always-possible-to-extend-a-ring-to-a-unital-ring?noredirect=1&lq=1 – Easy Aug 15 '18 at 05:10
  • @Chickenmancer I think we should note that a ring and its subring do not share the same "1" in general. – Easy Aug 15 '18 at 05:14
  • https://math.stackexchange.com/questions/170953/nontrivial-subring-with-unity-different-from-the-whole-ring

    They do share the same 1 in general.

     – Chickenmancer Aug 15 '18 at 06:32
  • @Chickenmancer Can you tell what are the "1"s in ${0}$ and $\mathbb{Z}$ respectively? – Easy Aug 15 '18 at 07:31

1 Answers1

7

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?

rschwieb
  • 153,510