Is there an example of a non-zero ring $R$ such that its proper subrings have identity but $R$ has no identity ? or even is it possible that proper non-zero subrings of $R$ have the same identity but $R$ has no identity element ??

closed as off topic by S. Carnahan♦Feb 9 '13 at 7:53

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– Theo Johnson-FreydFeb 9 '13 at 7:17

I am new in this site. I have revised the question.
– vitimiFeb 9 '13 at 7:45

Depending on your definition of "non-zero ring", there is a suitable object with 2 or 4 elements.
– S. Carnahan♦Feb 9 '13 at 7:55

In representation theory of $p$-adic groups, one of the fundamental constructions there is of the Hecke algebra, which is the space of locally constant functions (on the group) with compact support. If you put a Haar measure on the group then this space becomes a ring, typically with no identity. However the Hecke algebra is a union of sub-algebras consisting of functions which are bi-invariant under some fixed open compact $K$, and these bi-invariant functions form a subring with identity (the identity being the characteristic function of $K$ multiplied by a suitable constant).
– user30035Feb 9 '13 at 7:57