Does Subring Inherit Same Multiplicative Identity?

[boombaby]

should the subring inherit the same multiplicative identity of the original ring? assuming multiplicative identity is required in the definition of ring.

according to the book (Rotman's), $$1\in S$$ is required. But, does it mean S contains the same multiplicative identity, or contains its own multiplicative identity?

Consider Z_6 defined on Z by taking mod 6. And its subset S={3k| k is integers} equipped with the same operations as in Z_6.
S is itself a ring, having multiplicative identity 3. I'm wondering if S is called a subring of Z_6?

 

[morphism]

It's a matter of convention. Most authors, when they require rings to have a multiplicative identity, will stipulate that a subring must have the same multiplicative identity as the ring.

 

[boombaby]

Thanks a lot!