Can R be a subring with identity different from 1_S?

[quasar987]

[SOLVED] Identity in a subring

Homework Statement

In Dummit & Foote on the section on tensor product of modules (10.4 pp.359), the authors write

"Suppose that the ring R is a subring of the ring S. Throughout this section, we always assume that $$1_R=1_S$$ (this ensures that S is a unital R-module)."

Now, I just want to make sure I'm not missing something. Can R be a subring with identity whose identity 1_R is different from 1_S?

I would say "no" because S, together with its multiplication operation, forms a monoid... and in a monoid M, the identity e is the only element with the ability to do me=em=m for any and all m in M. So if 1_R were an identity for R different than 1_S, it would mean 1_R*r = 1_S*r = r and we would have two different element with the ability to act on the elements of r like an identity, which contradicts the fact that S is a (multiplicative) monoid.

So instead, perhaps a less confusing way to write the above quoted passage would be to say,

"Suppose that the ring R is a subring of the ring S. Throughout this section, we always assume that R contains $$1_S$$ (this ensures that S is a unital R-module)."

Yes? Thanks for the feedback.

 

[HallsofIvy]

You are clearly assuming that any ring has a multiplicative identity. Is Dummit and Foote assuming that?

 

[Dick]

Take S to be the ring of 2x2 matrices and R to be the subring consisting consisting of matrices [[x,0],[0,0]] for any x. [[1,0],[0,0]] is a unit for R, but not for S.