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[SOLVED] Identity in a subring
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 [tex]1_R=1_S[/tex] (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 [tex]1_S[/tex] (this ensures that S is a unital R-module)."
Yes? Thanks for the feedback.
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 [tex]1_R=1_S[/tex] (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 [tex]1_S[/tex] (this ensures that S is a unital R-module)."
Yes? Thanks for the feedback.