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Prove:

a) A subring $S$ of a ring with unity $R$ does not have to be a ring with unity.

b) A subring $S$ of a ring $R$ without unity can contain a unity.

c) A subring $S$ of a ring $R$ can have a unity which is different than $1_R$

In b) and c), is it enough to say that {0} is a subring of every ring, and 0 is a unity in it?

Also, how to show a)?

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    $\begingroup$ It should be made clear that this question and the answers below are based on a formulation of the notion of ring that does not require a ring to have a multiplicative unit. For many of us the term "ring" with no further qualification means "ring with unit" (and ring homomorphisms are required to preserve that unit) which leads to different answers to these questions. $\endgroup$
    – Rob Arthan
    Mar 3, 2021 at 22:22

2 Answers 2

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For part $a$, look at $2\mathbb{Z} \subseteq \mathbb{Z}$.

Your solution to parts $b$ and $c$ seem correct.

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a) You already know several subsets of $\Bbb{Z}$ which are closed under addition and multiplication and do not contain $1$. They're kernels of $\Bbb{Z} \rightarrow \Bbb{Z}: z \mapsto [z \pmod{p}]$ for primes $p$.

b) and c) If your definition of subring allows the trivial ring (which is usually the case), then the trivial ring, which you recite, is an example.

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